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Probability, Statistics & Linear Programming

Probability

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1. Axioms, conditional probability, independence, Bayes

Read this as a chapter →

Lec-03 Introduction to Probability

NPTEL · Probability and Statistics

Lec-04 Laws of Probability - I

NPTEL · Probability and Statistics

Conditional probability, independence and Bayes.

Probability axioms, conditioning and independence free

In plain words

Probability shares one unit of certainty among outcomes, and conditioning discards what is ruled out and rescales the rest. Keep the base rate in view, or a rare condition will mislead you.

Basics

P(A ∪ B) = P(A) + P(B) − P(A∩B); inclusion–exclusion for more sets. Boole: P(∪Ai)≤∑P(Ai)P(\cup{}A_{i}) \le \sum{}P(A_{i}). Bonferroni: P(∩Ai)≥1−∑P(AiP(\cap{}A_{i}) \ge 1 - \sum{}P(A_{i}ᶜ).

Conditioning

P(A|B) = P(A∩B)/P(B). Bayes: P(AiP(A_{i}|B) = P(B|Ai)P(Ai)/∑P(BA_{i})P(A_{i})/\sum{}P(B|Aj)P(Aj)A_{j})P(A_{j}).

Symmetry facts (the recurring Part-C pattern):

Statement Symmetric in A, B?
P(A|B) = 0 ⇔ P(B|A) = 0 ✓ (both say P(A∩B) = 0)
P(A|B) > P(A) ⇔ P(B|A) > P(B) ✓ (both say P(A∩B) > P(A)P(B))
P(A|B) = 1 ⇒ P(B|A) = 1 ✗ (A ⊇ B does not give B ⊇ A)
P(A|B) > P(B) ⇒ P(B|A) > P(A) ✗

Independence

  • A ⫫ B ⇔ P(A∩B) = P(A)P(B). Then Aᶜ ⫫ B too.
  • Pairwise independence does not imply mutual independence: two fair coins with A = first head, B = second head, C = "same face" are pairwise independent but not mutually.
  • Independent events with positive probability are never disjoint.

Standard traps

  • Mutually exclusive ≠ independent (they are opposites unless one has probability 0).
  • Conditional independence given C does not imply independence, and vice versa (Simpson's paradox territory).

Key takeaways

  • Convert every conditional statement to P(A∩B) vs P(A)P(B) and symmetry becomes obvious.
  • Pairwise ⇏ mutual independence.
  • Disjoint and independent are almost opposite notions.

Bayes and the base rate: why a 99% accurate test is usually wronginteractivefree

Step 1 / 4The setup

A test has sensitivity 0.990.99 and specificity 0.950.95; the disease affects 1%1\% of people. You test positive. What is the chance you have it?

Exam focus: Most Part-C items here are symmetry checks: P(A|B) > P(A) ⇔ P(A∩B) > P(A)P(B) is symmetric in A and B, while P(A|B) > P(B) is not.

Practice

Part C2025 Decembershow ▾
Suppose two fair dice are thrown independently at random. Let X and Y be the numbers on the upper face of the first die and that of the second die, respectively. Then which of the following statements are true?
  1. A.P(X−Y=0 ∣ X+Y=2) = P(X−Y=0 ∣ X+Y=12)✓ Correct
  2. B.E((X−Y)/(X+Y)) = 0✓ Correct
  3. C.Cov(X+Y, X−Y) = 0✓ Correct
  4. D.(X+Y) and (X−Y) are independent

The worked solution and the per-option analysis are in Trap Analysis.

Part B2025 Juneshow ▾
Suppose the distribution of X given θ\theta is normal with mean θ\theta and variance 15. Further, let the prior (improper) distribution of θ\theta be proportional to 1,−∞<θ<∞1, -\infty < \theta < \infty. If the observed value of X is 13, then which of the following statements is true?
  1. A.Posterior mean = Maximum likelihood estimate of θ\theta, Posterior variance = Var(X|θ)\theta)✓ Correct
  2. B.Posterior mean = Maximum likelihood estimate of θ\theta, Posterior variance < Var(X|θ)\theta)
  3. C.Posterior mean > Maximum likelihood estimate of θ\theta, Posterior variance = Var(X|θ)\theta)
  4. D.Posterior mean > Maximum likelihood estimate of θ\theta, Posterior variance < Var(X|θ)\theta)

The worked solution and the per-option analysis are in Trap Analysis.

Part C2025 Juneshow ▾
Let X1,X2,…,XnX_{1}, X_{2}, \dots, X_{n} be a random sample from N(θ,1),θ∈RN(\theta, 1), \theta \in \mathbb{R}. If θ\thetâ is the Bayes estimator of θ\theta with respect to some prior π(θ)\pi(\theta) and loss function L(θ,d)L(\theta, d). Then, which of the following statements are true?
  1. A.θ\thetâ =∑i= \sum_{i}₌1nXi/(n+τ2)_{1}^{n} X_{i}/(n + \tau^{2}), if the prior is N(0,1/τ2),τ2N(0, 1/\tau^{2}), \tau^{2} known and L(θ,d)=(θ−d)2L(\theta, d) = (\theta - d)^{2}✓ Correct
  2. B.θ\thetâ =∑i= \sum_{i}₌1nXi/(n+τ2)_{1}^{n} X_{i}/(n + \tau^{2}), if the prior is N(0,τ2),τ2N(0, \tau^{2}), \tau^{2} known and L(θ,d)=L(\theta, d) = |θ−d\theta - d|
  3. C.θ\thetâ =∑i= \sum_{i}₌1nXi/(n+1/τ2)_{1}^{n} X_{i}/(n + 1/\tau^{2}), if the prior is N(0,1/τ2),τ2N(0, 1/\tau^{2}), \tau^{2} known and L(θ,d)=L(\theta, d) = |θ−d\theta - d|
  4. D.θ\thetâ =∑i= \sum_{i}₌1nXi/n_{1}^{n} X_{i}/n, if the prior is the Jeffreys prior and L(θ,d)=(θ−d)2L(\theta, d) = (\theta - d)^{2}✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part C2025 Decembershow ▾
Suppose X∣θ\theta ~ Binomial(7,θ),0<θ<1(7,\theta), 0<\theta<1, and the prior distribution of θ\theta is Beta(α,β)(\alpha,\beta) where α>0\alpha>0 and β>0\beta>0 are known. Then which of the following statements MAY NOT be true?
  1. A.Posterior mean of θ\theta given X=2 is less than the prior mean✓ Correct
  2. B.Posterior mean of θ\theta given X=3 is greater than the prior mean✓ Correct
  3. C.Posterior mean of θ\theta given X=4 is not equal to the prior mean✓ Correct
  4. D.Posterior mean of θ\theta given X=5 is equal to the prior mean✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part B2025 Juneshow ▾
A biased six-faced die is tossed once. Suppose that the probability of any prime number showing up is twice that of any non-prime number showing up. Then, the probability that an odd number will show up is
  1. A.1/3
  2. B.2/3
  3. C.4/9
  4. D.5/9✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part B2024 Juneshow ▾
Let A1,A2,A3A_{1}, A_{2}, A_{3} be events satisfying 0<P(Ai)<10 < P(A_{i}) < 1 for i = 1, 2, 3. Which of the following statements is true?
  1. A.P(A1P(A_{1} | A2)P(A2A_{2})P(A_{2} | A3)≤P(A1A_{3}) \le P(A_{1} | A3)A_{3})
  2. B.P(A1P(A_{1} | A2)P(A3A_{2})P(A_{3} | A2)≥P(A1∩A3A_{2}) \ge P(A_{1} \cap A_{3} | A2)A_{2})
  3. C.P(A1P(A_{1} | A2)+P(A3A_{2}) + P(A_{3} | A2)≥P(A1∪A3A_{2}) \ge P(A_{1} \cup A_{3} | A2)A_{2})✓ Correct
  4. D.P(A1P(A_{1} | A2)+P(A2A_{2}) + P(A_{2} | A3)≤P(A1A_{3}) \le P(A_{1} | A3)A_{3})

The worked solution and the per-option analysis are in Trap Analysis.

Part B2024 Decembershow ▾
Let U1,U2,…,U5U_{1}, U_{2}, \dots, U_{5} be 5 urns such that urn U_k contains 2k+k22k + k^{2} balls, out of which 2k are white balls and k2k^{2} are black balls, k = 1, 2, …, 5. An urn is selected with probability of selecting urn U_k being proportional to (k + 2). A ball is chosen randomly from the selected urn. Then, the probability that the urn U5U_{5} was selected, given that the ball drawn is white, is equal to
  1. A.3/5
  2. B.2/5
  3. C.1/5✓ Correct
  4. D.3/4

The worked solution and the per-option analysis are in Trap Analysis.

Part B2024 Decembershow ▾
Let U1,U2,…,U6U_{1}, U_{2}, \dots, U_{6} be 6 urns such that urn U_k contains 3k+k23k + k^{2} balls, out of which 3k are white balls and k2k^{2} are black balls, k = 1, 2, …, 6. An urn is selected with the probability of selecting urn U_k being proportional to (k + 3). A ball is chosen randomly from the selected urn. Then the probability that urn U6U_{6} was selected, given that the ball drawn is white, is equal to
  1. A.7/13
  2. B.6/13
  3. C.1/6✓ Correct
  4. D.7/9

The worked solution and the per-option analysis are in Trap Analysis.

Part C2023 Juneshow ▾
Let A, B be two events in a discrete probability space with P(A) > 0 and P(B) > 0. Which of the following are necessarily true?
  1. A.If P(A | B) = 0 then P(B | A) = 0.✓ Correct
  2. B.If P(A | B) = 1 then P(B | A) = 1.
  3. C.If P(A | B) > P(A) then P(B | A) > P(B).✓ Correct
  4. D.If P(A | B) > P(B) then P(B | A) > P(A).

Solution

(1) P(A∩B) = 0 is symmetric.

(3) P(A|B) > P(A) ⇔ P(A∩B) > P(A)P(B), symmetric in A, B.

(2) A ⊇ B (a.s.) does not give B ⊇ A.

(4) Take B ⊂ A with P(B) small.

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Part Cshow ▾
Roll a fair die once; let A = {even}, B = {1, 2}, C = {1, 2, 3, 4}. Which of the following are true?
  1. A.A and B are independent.✓ Correct
  2. B.A and C are independent.✓ Correct
  3. C.P(A | B) = 1/2.✓ Correct
  4. D.Pairwise independence of three events implies their mutual independence.

The worked solution and the per-option analysis are in Trap Analysis.

Part Cshow ▾
A test for a disease has sensitivity 0.99 and specificity 0.95, and the disease affects 1% of the population. Which of the following are true?
  1. A.P(test positive) = 0.0594.✓ Correct
  2. B.P(diseased | positive) = 1/6.✓ Correct
  3. C.P(diseased | positive) > 1/2.
  4. D.Halving the prevalence to 0.5% lowers P(diseased | positive) further.✓ Correct

Solution

P(+) = 0.99×0.01 + 0.05×0.99 = 0.0099 + 0.0495 = 0.0594.

Bayes: P(D|+) = 0.0099/0.0594 = 1/6 ≈ 0.17, nowhere near 1/2 — the base-rate fallacy in numbers.

Lower prevalence shrinks the true-positive share of the positives, dropping the posterior further.

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Part Bshow ▾
Let P(A) = 0.6 and P(B) = 0.5. The smallest possible value of P(A ∩ B) is
  1. A.0
  2. B.0.1✓ Correct
  3. C.0.3
  4. D.0.5

The worked solution and the per-option analysis are in Trap Analysis.

2. Random variables, distributions, moments, MGF

Read this as a chapter →

Lec-07 Random Variables

NPTEL · Probability and Statistics

Lec-08 Probability Distributions

NPTEL · Probability and Statistics

Lec-09 Characteristics of Distributions

NPTEL · Probability and Statistics

Moments and generating functions.

Random variables, moments and generating functions free

In plain words

The cumulative function always exists. The mean does not. Cauchy is the standing warning: an ordinary-looking bell shape whose tails are too heavy for a mean to exist at all.

Moments

E[X]=∫xE[X] = \int{}x dF; Var=E[X2]−(E[X])2Var = E[X^{2}] - (E[X])^{2}. Existence can fail: the Cauchy distribution has no mean, and tνt_\nu has moments only up to order ν−1\nu - 1.

Markov: P(X ≥ a) ≤ E[X]/a for X ≥ 0. Chebyshev: P(|X−μX - \mu| ≥kσ)≤1/k2\ge k\sigma) \le 1/k^{2}. Jensen: φ\varphi convex ⇒φ(EX)≤E[φ(X)]\Rightarrow \varphi(E X) \le E[\varphi(X)].

MGF and characteristic function

M(t) = E[etX],φ(t)=E[eitXe^{tX}], \varphi(t) = E[e^{itX}].

  • M determines the distribution when it exists in a neighbourhood of 0;φ0; \varphi always exists and always determines it.
  • Independent sums: MX+YM_{X+Y} = M_X M_Y. This is how you recognise convolutions.
  • M⁽n^{n}⁾(0)=E[Xn](0) = E[X^{n}].
  • Cauchy has no MGF (and no mean); its characteristic function is e−∣t∣e^{-|t|}.

Recognising from the MGF

M(t) Distribution
(q + pet)n^t)^{n} Binomial(n, p)
eλ(et−1)e^{\lambda(e^t - 1)} Poisson(λ)(\lambda)
(1 − t/λ)−kt/\lambda)^{-k} Gamma(k,λ)(k, \lambda)
eμt+σ2t2/2e^{\mu{}t + \sigma^{2}t^{2}/2} N(μ,σ2)N(\mu, \sigma^{2})
p e^t/(1 − qe^t) Geometric

A product of MGFs of different families means independence and a sum, not a mixture. Mixtures add densities: f=αf1+(1−α)f2f = \alpha{}f_{1} + (1-\alpha)f_{2} gives E[Y]=αE[X1]+(1−α)E[X2]E[Y] = \alpha{}E[X_{1}] + (1-\alpha)E[X_{2}] but Y≠αX1+(1−α)X2Y \ne \alpha{}X_{1} + (1-\alpha)X_{2}.

Key takeaways

  • MGF factorises for sums; densities combine for mixtures — do not confuse them.
  • Cauchy: no mean, no MGF, but a perfectly good characteristic function.
  • Chebyshev and Markov are the standard tools for "which bound is guaranteed" questions.

Moments, MGFs, and when they fail to existinteractive

This visual is part of Trap Analysis

The tail-integral formula, the two inequalities worth memorising, and a distribution with no mean at all.

Unlock interactive visuals

Exam focus: MGF determines the distribution and factorises over independent sums. Know which distributions have no MGF (Cauchy, t) and which moments fail to exist.

Practice

Part C2025 Juneshow ▾
Let X be a random variable with the cumulative distribution function F_X(x) = 0 if x < 0, F_X(x) = (x + 2)/5 if 0 ≤ x < 2, and F_X(x) = 1 if x ≥ 2. Then, which of the following statements are true?
  1. A.E(X2)=4/3E(X^{2}) = 4/3✓ Correct
  2. B.P(|X − 4/5| ≥ 1) = 19/25
  3. C.The upper bound of P(|X − 4/5| ≥ 1), using Chebyshev's inequality, is 52/75✓ Correct
  4. D.The value of the moment generating function, M_X(t), at t = 1 is (e2−1)/5(e^{2} - 1)/5

The worked solution and the per-option analysis are in Trap Analysis.

Part C2025 Decembershow ▾
Let X and Y be two independent random variables such that the moment generating functions of X and Y are MX(t)=e(3(et−1)),t∈RM_X(t)=e^(3(e^t-1)), t\in\mathbb{R}, and MY(t)=((1/2)e(−3t)+(1/2)e(3t))2,t∈RM_Y(t)=((1/2)e^(-3t)+(1/2)e^(3t))^{2}, t\in\mathbb{R}, respectively. Then which of the following statements are true?
  1. A.P(XY=0) = (1+e⁻3)/2^{3})/2✓ Correct
  2. B.E(X+Y) = 3✓ Correct
  3. C.Var(X+Y) = 21✓ Correct
  4. D.Cov(X+Y,X−Y) = 0

The worked solution and the per-option analysis are in Trap Analysis.

Part B2025 Decembershow ▾
Let {Yn:n≥1Y_{n} : n \ge 1} be a sequence of independent and identically distributed random variables, where Y1Y_{1} ~ Bernoulli(1/2). Define Z=∑n=1∞Z = \sum_{n=1}^\infty 4Yn/5n4Y_{n}/5^{n}. Then, which of the following statements is true?
  1. A.P(Z ≥ 3/5) = 0.5, P(Z = 4/25) = 0✓ Correct
  2. B.P(Z ≥ 3/5) = 0.6, P(Z = 4/25) = 0
  3. C.P(Z ≥ 3/5) = 0.7, P(Z = 4/25) = 0.16
  4. D.P(Z ≥ 3/5) = 0.8, P(Z = 4/25) = 0.16

The worked solution and the per-option analysis are in Trap Analysis.

Part B2025 Juneshow ▾
Suppose that we have a data set consisting of 2n + 1 observations for some n∈Nn \in \mathbb{N}. Value of each observation is either x or x + r, where x∈N,r≥0x \in \mathbb{N}, r \ge 0. Then, which of the following statements is always true?
  1. A.The mean and median of the data will be different if and only if r > 0
  2. B.Variance of the data is positive if and only if r > 0
  3. C.Mean and mode of the data will be same if and only if r = 0
  4. D.Median and mode of the data will be same for all values of r ≥ 0✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part B2024 Juneshow ▾
Let X be a random variable with cumulative distribution function given by F(x) = 0 if x < 0; F(x) = (x + 1)/3 if 0 ≤ x < 1; F(x) = 1 if x ≥ 1. Then the value of P(1/3 < X < 3/4) + P(X = 0) is equal to
  1. A.7/36
  2. B.11/36
  3. C.13/36
  4. D.17/36✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Decembershow ▾
Let X and Y be independent random variables with the moment generating functions M_X(t) = (1/9)(2 + eᵗ)2,t∈R)^{2}, t \in \mathbb{R} and M_Y(t) = exp[eᵗ −1],t∈R- 1], t \in \mathbb{R}, respectively. Then, which of the following statements are true?
  1. A.P(XY = 0) = (4 + 5e⁻1)/9^{1})/9✓ Correct
  2. B.E[(3X−Y)2]=6E[(3X - Y)^{2}] = 6✓ Correct
  3. C.Var(X + Y) = 2
  4. D.Cov(2X + Y, X − 2Y) = −10/9✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part B2024 Decembershow ▾
Suppose X ~ Uniform(5, 10). Define Z = X + 3 if X ≤ 7, and Z = X − 3 otherwise. Then E(Z) is
  1. A.4.5
  2. B.6.9✓ Correct
  3. C.7.5
  4. D.34.5

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Decembershow ▾
Let X be a random variable with probability density function f(x)=1/(π(1+x2)),x∈Rf(x) = 1/(\pi(1 + x^{2})), x \in \mathbb{R}. If Z=1/2+(1/π)tan⁡Z = 1/2 + (1/\pi)\tan⁻1(X)^{1}(X), then which of the following statements are true?
  1. A.E(Zᵐ) = 1/(m + 1), for all m∈Nm \in \mathbb{N}✓ Correct
  2. B.E(ΦE(\Phi⁻1(Z))=0^{1}(Z)) = 0, where Φ(⋅)\Phi(\cdot) is the cumulative distribution function of standard normal random variable.✓ Correct
  3. C.Z is degenerate at 0.
  4. D.If Z1Z_{1} and Z2Z_{2} are independent and identically distributed (i.i.d.) random variables having distribution same as the distribution of Z, then Z has the same distribution as (Z1+Z2)/2(Z_{1} + Z_{2})/2.

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Juneshow ▾
Let X and Y be independent random variables with X ~ N(2, 4) and Y ~ N(−4, 9) where N(μ,σ2)N(\mu, \sigma^{2}) denotes a normal distribution with mean μ\mu and variance σ2\sigma^{2}. Given Φ(1)=0.8413,Φ(2)=0.9772\Phi(1) = 0.8413, \Phi(2) = 0.9772 and Φ(3)=0.9987\Phi(3) = 0.9987 where Φ(⋅)\Phi(\cdot) is the cumulative distribution function of a standard normal random variable. Which of the following statements are true?
  1. A.Var(2X + Y) = 17
  2. B.P(|2X + Y| ≤ 15) = 0.9974✓ Correct
  3. C.Cov(3X + 2Y, 3X − 2Y) = 0✓ Correct
  4. D.2X − Y ~ N(0, 25)

The worked solution and the per-option analysis are in Trap Analysis.

Part C2023 Decembershow ▾
Let X be a discrete random variable with support {0, 1, …, 25} and P(X=x)=C(25,x)/225P(X = x) = C(25, x)/2^{25}. Then which of the following statements are true?
  1. A.The distributions of X − 12.5 and 12.5 − X are identical✓ Correct
  2. B.P(X ≤ 4) = P(X ≥ 22)
  3. C.The coefficient of variation (in percentage) of X is 20✓ Correct
  4. D.P(X ≤ 4.9) = P(X ≥ 20.1)✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part C2023 Decembershow ▾
Suppose U ~ Uniform(0,1) and X=tan⁡(π(U−X = \tan(\pi(U - ½)). Then which of the following statements are true?
  1. A.E(X4)=3E(X^{4}) = 3
  2. B.P(X ∈ {1, 2, 5}) = 1/2
  3. C.E(eˣ) does not exist✓ Correct
  4. D.P(X ≤ 0) = 1/2✓ Correct

Solution

X is standard Cauchy: symmetric about 0, continuous (so any finite set has probability 0), and with no finite moments — E(eˣ) diverges too.

Save for spaced revision (free account)
Part C2023 Decembershow ▾
Suppose X is a continuous random variable with density f(x)=(1/π)⋅1/(1+(x+1)2)f(x) = (1/\pi)\cdot1/(1 + (x + 1)^{2}), and Y = X/|X| for X ≠ 0, Y = 0 for X = 0. Then which of the following statements are true?
  1. A.E(Y) = 0
  2. B.P(Y > 0) < P(Y < 0)✓ Correct
  3. C.P(Y < −1) < P(Y > 1)
  4. D.E(Y2)=1E(Y^{2}) = 1✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part B2023 Decembershow ▾
Let (X, Y) have joint moment generating function M(t1,t2)=(3/4+M(t_{1}, t_{2}) = (3/4 + et1/4)2e^{t_{1}}/4)^{2} (5/6 + et2/6)3e^{t_{2}}/6)^{3}. Then P(X + 2Y > 1) equals
  1. A.1581/3456✓ Correct
  2. B.1875/3456
  3. C.125/3456
  4. D.3331/3456

The worked solution and the per-option analysis are in Trap Analysis.

Part B2023 Decembershow ▾
Suppose X ~ Poisson(3/4). Then which of the following statements is true?
  1. A.P(X > 9) ≥ 11/12
  2. B.P(X < 9) ≥ 11/12✓ Correct
  3. C.E(X−3/4)2≥11/12E(X - 3/4)^{2} \ge 11/12
  4. D.(11/9)X ~ Poisson(11/12)

Solution

Markov: P(X ≥ 9) ≤ E[X]/9 = 1/12, so P(X < 9) ≥ 11/12.

The variance is 3/4 < 11/12, and a scaled Poisson is not Poisson.

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Part C2023 Decembershow ▾
Let X be a discrete random variable with support S = {−1, 0, 1} and P(X = 0) = 1/3. Then which of the following statements are true?
  1. A.E(X) ≤ 2/3✓ Correct
  2. B.E(X2)=2/3E(X^{2}) = 2/3✓ Correct
  3. C.E(|X|) = 2/3✓ Correct
  4. D.Var(X) > 2/3

The worked solution and the per-option analysis are in Trap Analysis.

Part Cshow ▾
Let X ≥ 0 be a continuous random variable with finite mean. Which of the following are true?
  1. A.E[X]=∫0∞P(X>t)E[X] = \int_{0}^\infty P(X > t) dt.✓ Correct
  2. B.P(X ≥ a) ≤ E[X]/a for every a > 0.✓ Correct
  3. C.If E[X2]<∞E[X^{2}] < \infty, then P(|X − EX| ≥k)≤Var(X)/k2\ge k) \le Var(X)/k^{2} for every k > 0.✓ Correct
  4. D.Finite mean implies finite variance.

The worked solution and the per-option analysis are in Trap Analysis.

Part Bshow ▾
Let X ~ Poisson(2). Then E[X2]E[X^{2}] equals
  1. A.2
  2. B.4
  3. C.6✓ Correct
  4. D.8

Solution

E[X2]=VarX+(EX)2=2+4=6E[X^{2}] = Var X + (E X)^{2} = 2 + 4 = 6, using the Poisson signature Var = mean =λ= \lambda.

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3. Standard discrete and continuous distributions

Read this as a chapter →

Lec-10 Special Distributions - I

NPTEL · Probability and Statistics

The standard families; keep the mean/variance table beside you.

Lec-11 Special Distributions - II

NPTEL · Probability and Statistics

The standard distributions free

In plain words

These are a family tree rather than a list to memorise. A binomial is a sum of coin flips, a Poisson is a binomial with rare trials, a gamma is a sum of exponentials.

Reference table

Distribution Mean Variance
Bernoulli(p) p pq
Binomial(n,p) np npq
Poisson(λ)(\lambda) λ\lambda λ\lambda
Geometric (trials, p) 1/p q/p2q/p^{2}
Negative binomial(r,p) r/p rq/p2/p^{2}
Hypergeometric nK/N nK(N−K)(N−n)/(N2(N−1))(N-K)(N-n)/(N^{2}(N-1))
Uniform(a,b) (a+b)/2 (b−a)2/12(b-a)^{2}/12
Exponential(λ)(\lambda) 1/λ1/\lambda 1/λ21/\lambda^{2}
Gamma(k,λ)(k,\lambda) k/λk/\lambda k/λ2k/\lambda^{2}
Beta(a,b) a/(a+b) ab/((a+b)2(a+b+1))/((a+b)^{2}(a+b+1))
N(μ,σ2)N(\mu,\sigma^{2}) μ\mu σ2\sigma^{2}
χn2\chi^{2}_{n} n 2n
tνt_\nu 0(ν>1)0 (\nu>1) ν/(ν−2)(ν>2)\nu/(\nu-2) (\nu>2)
Cauchy — —

Relationships to quote

  • Sum of n i.i.d. Exp(λ)=(\lambda) = Gamma(n,λ)(n, \lambda); Gamma(n/2, ½)=χn2) = \chi^{2}_{n}.
  • min of independent Exp(λi)(\lambda_{i}) ~ Exp(∑λi)(\sum\lambda_{i}) — the series-system fact.
  • Sum of independent Poissons is Poisson; of independent normals is normal; of independent χ2\chi^{2} is χ2\chi^{2}.
  • Z ~ N(0,1)⇒Z2N(0,1) \Rightarrow Z^{2} ~ χ12.t=Z/χn2/n;F=(χm2/m)/(χn2/n)\chi^{2}_{1}. t = Z/\sqrt{\chi^{2}_{n}/n}; F = (\chi^{2}_{m}/m)/(\chi^{2}_{n}/n).
  • Binomial → Poisson (n large, np fixed) → Normal (CLT).
  • Memorylessness characterises the exponential (continuous) and geometric (discrete): P(X > s+t | X > s) = P(X > t).

Order statistics

For i.i.d. with CDF F:P(X(n)≤x)=F(x)nF: P(X_{(n)} \le x) = F(x)^{n} and P(X(1)>x)=(1−F(x))nP(X_{(1)} > x) = (1 - F(x))^{n}. For Uniform(0,1),X(k)(0,1), X_{(k)} ~ Beta(k, n−k+1).

Key takeaways

  • Exponential: memoryless, minimum stays exponential, sum becomes gamma. −χ2,t- \chi^{2}, t and F are all built from normals — know the constructions.
  • Uniform order statistics are Beta.

The signatures that identify a distribution instantlyinteractive

This visual is part of Trap Analysis

One characterising property each for Poisson, exponential and normal.

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A distribution's own shape, as its parameters moveinteractivefree

Exam focus: Know the mean/variance table cold and the standard relationships (sum of exponentials = gamma, min of exponentials = exponential, memorylessness).

Practice

Part C2025 Decembershow ▾
Let {Xn:n≥1X_{n} : n\ge1} be a sequence of independent and identically distributed random variables, where X1X_{1} has an Exponential distribution with mean 1. Define Tn=max⁡T_{n} = \max{X1,X2,…,XnX_{1},X_{2},\dots,X_{n}} − ln n, n≥1. Suppose Tn→T_{n} \toᵈ Y as n→∞n\to\infty. Then which of the following statements are true?
  1. A.Y has a Double Exponential distribution with location parameter ln(ln 2) and scale parameter 1
  2. B.Median of Y = ln(ln 2)
  3. C.P(Y≤0) = e⁻1^{1}✓ Correct
  4. D.The derivative of the cumulative distribution function of Y at ln 3 is e^(−1/3)

The worked solution and the per-option analysis are in Trap Analysis.

Part C2025 Juneshow ▾
Let X and Y be independent Poisson random variables with means 4 and 2, respectively. Then, which of the following statements are true?
  1. A.The conditional distribution of X given X + Y = 3 is Binomial(3, 1/3)
  2. B.P(X ≤ 1 | X + Y = 3) = 7/27✓ Correct
  3. C.E(X | X + Y = 3) = 2✓ Correct
  4. D.The value of the characteristic function of X + Y at the point t=πt = \pi is e−12e^{-12}✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part B2025 Juneshow ▾
A mobile manufacturing company uses two brands of batteries for its mobiles. The life (in years) of batteries of Brand I follows an exponential distribution with the probability density function f(x) = e−xe^{-x} if x > 0 and 0 otherwise, and that of Brand II follows a gamma distribution with the probability density function g(x) = (x/4)e−x/2(x/4)e^{-x/2} if x > 0 and 0 otherwise. The company uses the batteries of Brands I and II in proportion of 20% and 80% respectively, in its mobiles. The probability that a randomly selected mobile has the battery life more that 2 years is
  1. A.(13/5)e−2(13/5)e^{-2}
  2. B.(1/5)(e−2(1/5)(e^{-2} + 2e−1)2e^{-1})
  3. C.(1/5)(e−2(1/5)(e^{-2} + 8e−1)8e^{-1})✓ Correct
  4. D.(1/5)(4e−2(1/5)(4e^{-2} + 2e−1)2e^{-1})

The worked solution and the per-option analysis are in Trap Analysis.

Part C2025 Juneshow ▾
A system has two components C1C_{1} and C2C_{2} put in parallel. The components C1C_{1} and C2C_{2} have independent lifetimes X1X_{1} and X2X_{2}, respectively. The probability distribution of X_j is exponential with mean 1/j, j = 1, 2. Suppose that R(t) and h(t) are the reliability and the hazard rate functions of the system, respectively. Then, which of the following statements are true?
  1. A.R(t) = e−2te^{-2t} + e−te^{-t} − e−3te^{-3t}, t > 0✓ Correct
  2. B.The expected lifetime of the given system is 1
  3. C.h(1)=(2e+e2−3)/(e+e2−1)h(1) = (2e + e^{2} - 3)/(e + e^{2} - 1)✓ Correct
  4. D.h(3)=(4e3+e6−5)/(e3+e6−1)h(3) = (4e^{3} + e^{6} - 5)/(e^{3} + e^{6} - 1)

The worked solution and the per-option analysis are in Trap Analysis.

Part C2025 Decembershow ▾
Consider a series system comprising four components, having independent and identically distributed lifetimes with hazard rate λ(t)=1/(1+t),t>0\lambda(t)=1/(1+t), t>0, and survival function S(t), t>0. If Y denotes the lifetime of the series system, then which of the following statements are true?
  1. A.Cumulative hazard function of each component is H(t)=2ln(1+t), t>0
  2. B.S(t)=1/(1+t)2,t>0S(t)=1/(1+t)^{2}, t>0
  3. C.P(Y<1/2)={S(1/2)}4^{4}
  4. D.P(Y<1/2)=65/81✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Juneshow ▾
Let X1,…,X12X_{1}, \dots, X_{12} be a random sample from the N(2, 4) distribution and Y1,…,Y15Y_{1}, \dots, Y_{15} be a random sample from the N(−2, 5) distribution, where N(μ,σ2)N(\mu, \sigma^{2}) denotes a normal distribution with mean μ\mu and variance σ2\sigma^{2}. Assume that the two random samples are mutually independent. Let Xˉ=(1/12)∑i{\bar{X}} = (1/12)\sum_{i}₌112Xi,S12=(1/11)∑i_{1}^{12}X_{i}, S_{1}^{2} = (1/11)\sum_{i}₌112(Xi−Xˉ)2_{1}^{12}(X_{i} - {\bar{X}})^{2}, Ȳ =(1/15)∑j= (1/15)\sum_{j}₌115Yj,S22=(1/14)∑j_{1}^{15}Y_{j}, S_{2}^{2} = (1/14)\sum_{j}₌115(Yj−_{1}^{15}(Y_{j} - Ȳ)2)^{2}. Which of the following statements are true?
  1. A.The distribution of X̄ + Ȳ is N(0, 2/3)✓ Correct
  2. B.The distribution of (1/20)(55S12+56S22)(1/20)(55S_{1}^{2} + 56S_{2}^{2}) is χ262\chi^{2}_{26}
  3. C.The distribution of (5/4)(S12/S22)(5/4)(S_{1}^{2}/S_{2}^{2}) is F11,14F_{11},_{14}✓ Correct
  4. D.The distribution of 23(2\sqrt{3}(Ȳ +2)/S1+ 2)/S_{1} is t14t_{14}

The worked solution and the per-option analysis are in Trap Analysis.

Part B2024 Decembershow ▾
Suppose that X ~ binomial(9,θ),0.7<θ<1(9, \theta), 0.7 < \theta < 1, and Y ~ Poisson(λ),λ>0(\lambda), \lambda > 0. If 3E(Y) = E(X), then which of the following is true?
  1. A.Var(X) > 3Var(Y)
  2. B.2Var(Y) < Var(X) < 3Var(Y)
  3. C.Var(Y) < Var(X) < 2Var(Y)
  4. D.Var(X) < Var(Y)✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part B2024 Juneshow ▾
Let X1,X2X_{1}, X_{2} be a random sample from N(0,σ2)N(0, \sigma^{2}) distribution, where σ>0\sigma > 0 and N(μ,σ2)N(\mu, \sigma^{2}) denotes a normal distribution with mean μ\mu and variance σ2\sigma^{2}. Suppose, for some constant c,(c(X12+X22),∞)c, (c(X_{1}^{2} + X_{2}^{2}), \infty) is a confidence interval for variance σ2\sigma^{2} with confidence coefficient 0.95. Then the value of c is equal to
  1. A.−2 ln(0.05)
  2. B.−2 ln(0.95)
  3. C.−1/(2 ln(0.05))✓ Correct
  4. D.−1/(2 ln(0.95))

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Decembershow ▾
Let Z1Z_{1} and Z2Z_{2} be two independent discrete random variables such that Z1Z_{1} follows binomial distribution with parameters n = 2 and p = 1/2, and Z2Z_{2} follows Poisson distribution with mean 1. Consider the following system of equations with three variables x1,x2x_{1}, x_{2} and x3:x1−2x2+x3=1;2x1−5x2+2x3=2;x1+2x2+Z1x3=Z2x_{3}: x_{1} - 2x_{2} + x_{3} = 1; 2x_{1} - 5x_{2} + 2x_{3} = 2; x_{1} + 2x_{2} + Z_{1}x_{3} = Z_{2}. Then, the probability that the given system of equations has infinite number of solutions equals
  1. A.e⁻1^{1}
  2. B.e⁻1/2^{1}/2✓ Correct
  3. C.e⁻1/4^{1}/4
  4. D.e⁻1/12^{1}/12

The worked solution and the per-option analysis are in Trap Analysis.

Part C2023 Juneshow ▾
Let X1X_{1} and X2X_{2} be independent, X1X_{1} gamma with mean 10 and variance 10, and X2X_{2} ~ N(3, 4). Let f1,f2f_{1}, f_{2} be their densities. Define Y with density f(y)=0.4f1(y)+qf2(y)f(y) = 0.4 f_{1}(y) + q f_{2}(y). Which of the following are true?
  1. A.q = 0.6✓ Correct
  2. B.E[Y] = 5.8✓ Correct
  3. C.Var(Y) = 3.04
  4. D.Y=0.4X1+Y = 0.4X_{1} + qX2_{2}

Solution

Densities integrate to 1 ⇒ q = 0.6.

Mixture mean 0.4·10 + 0.6·3 = 5.8.

E[Y2]=0.4(10+100)+0.6(4+9)=51.8E[Y^{2}] = 0.4(10 + 100) + 0.6(4 + 9) = 51.8, so Var(Y)=51.8−5.82=18.16Var(Y) = 51.8 - 5.8^{2} = 18.16.

A mixture is not a linear combination of the variables.

Save for spaced revision (free account)
Part B2023 Juneshow ▾
Let X1,X2,X3,X4X_{1}, X_{2}, X_{3}, X_{4} be i.i.d. Bernoulli(1/3) and X(1)≤⋯≤X(4)X_{(1)} \le \cdots \le X_{(4)} the order statistics. Which of the following is true?
  1. A.X(1)X_{(1)} and X(4)X_{(4)} are independent.
  2. B.Expectation of X(2)X_{(2)} is 1/2.
  3. C.Variance of X(2)X_{(2)} is 8/81.✓ Correct
  4. D.X(4)X_{(4)} is a degenerate random variable.

The worked solution and the per-option analysis are in Trap Analysis.

Part Cshow ▾
Let X ~ Exponential(λ)(\lambda), with density λe−λx\lambda{}e^{-\lambda{}x} on x > 0. Which of the following are true?
  1. A.P(X > s + t | X > s) = P(X > t) for all s, t > 0.✓ Correct
  2. B.E[X]=1/λE[X] = 1/\lambda and Var(X)=1/λ2Var(X) = 1/\lambda^{2}.✓ Correct
  3. C.For independent exponentials with rates λ1\lambda_{1} and λ2,min⁡(X1,X2)\lambda_{2}, \min(X_{1}, X_{2}) is exponential with rate λ1+λ2\lambda_{1} + \lambda_{2}.✓ Correct
  4. D.The exponential is the only continuous distribution on (0,∞)(0, \infty) with the property in option 1.✓ Correct

Solution

(1) Memorylessness: e^{−λ(s+t)-\lambda(s+t)}/e^{−λs-\lambda{}s} = e−λte^{-\lambda{}t}.

(2) Standard moments.

(3) P(min > t) = e−λ1te^{-\lambda_{1}t}·e−λ2te^{-\lambda_{2}t}.

(4) The survival function must satisfy G(s + t) = G(s)G(t), and the only monotone solutions are exponentials — the characterisation theorem.

All four hold.

Save for spaced revision (free account)
Part Bshow ▾
Let X ~ N(μ,σ2)N(\mu, \sigma^{2}). The probability P(|X−μX - \mu| <σ)< \sigma) is approximately
  1. A.0.50
  2. B.0.68✓ Correct
  3. C.0.90
  4. D.0.95

The worked solution and the per-option analysis are in Trap Analysis.

Part Cshow ▾
Let X follow a Poisson distribution with parameter λ>0\lambda > 0. Which of the following are true?
  1. A.E[X]=Var(X)=λE[X] = Var(X) = \lambda.✓ Correct
  2. B.The moment generating function of X is exp⁡(λ(e\exp(\lambda(eᵗ − 1)).✓ Correct
  3. C.X takes negative values with positive probability.
  4. D.The distribution of X is symmetric about its mean.

The worked solution and the per-option analysis are in Trap Analysis.

Part Bshow ▾
A fair coin is tossed repeatedly. The expected number of tosses needed to obtain the first head is
  1. A.1
  2. B.2✓ Correct
  3. C.3
  4. D.4

The worked solution and the per-option analysis are in Trap Analysis.

4. Joint distributions, transformations, order statistics

Read this as a chapter →

Lec-18 Joint Distributions - I

NPTEL · Probability and Statistics

Lec-22 Transformations of Random Vectors

NPTEL · Probability and Statistics

The Jacobian method for transformations.

Joint distributions, transformations and order statistics free

In plain words

Independence means the density genuinely factors. Uncorrelated is far weaker, because correlation sees only the straight-line part — a variable and its own square have correlation zero.

Independence vs correlation

X ⫫ Y ⇒ Cov = 0. The converse fails: X ~ N(0,1) and Y=X2Y = X^{2} are uncorrelated (E[X3]=0)(E[X^{3}] = 0) but dependent.

Exception: if (X, Y) is jointly normal, then uncorrelated ⇔ independent. Marginal normality is not enough — X ~ N(0,1) and Y=εXY = \varepsilon{}X with ε=±1\varepsilon = \pm1 independent of X are each normal, uncorrelated, and dependent.

Transformations

For (U,V) = g(X,Y) with a bijective smooth g: fU,V(uf_{U,V}(u,v) = fX,Y(xf_{X,Y}(x,y)|J| where J=∂(x,y)/∂(u,v)J = \partial(x,y)/\partial(u,v).

Standard results:

  • X, Y ~ N(0,1) independent ⇒ X/Y ~ Cauchy, X2+Y2X^{2} + Y^{2} ~ χ22=\chi^{2}_{2} = Exp(½).
  • X ~ U(0,1) ⇒ −ln X ~ Exp(1);tan⁡(π(X−(1); \tan(\pi(X - ½)) ~ Cauchy.
  • Probability integral transform: F(X) ~ U(0,1) for continuous F; F⁻1(U)^{1}(U) simulates any distribution.
  • Box–Muller: −2lnU1cos⁡(2πU2)\sqrt{-2ln U_{1}}\cos(2\pi{}U_{2}) ~ N(0,1).

Order statistics

fX(k)(x)f_{X_{(k)}}(x) = n!/((k−1)!(n−k)!) F(x)k−1(1F(x)^{k-1}(1 − F(x))n−kF(x))^{n-k} f(x).

  • Joint density of all order statistics: n!∏f(xi)n!\prod{}f(x_{i}) on the ordered region. −X(1)- X_{(1)} and X(n)X_{(n)} are not independent (they are positively associated).
  • For Uniform(0,1):E[X(k)]=k/(n+1)(0,1): E[X_{(k)}] = k/(n+1).
  • Range R=X(n)−X(1)R = X_{(n)} - X_{(1)}; for the uniform, R ~ Beta(n−1, 2).

Conditional expectation

E[X] = E[E[X|Y]] (tower); Var(X) = E[Var(X|Y)] + Var(E[X|Y]).

Key takeaways

  • Uncorrelated ⇒ independent only under joint normality.
  • Marginally normal ≠ jointly normal.
  • The tower property and variance decomposition solve most conditional questions.

Order statistics from first principlesinteractive

This visual is part of Trap Analysis

Derive the max, the min and the k-th — with the dependence that catches people out.

Unlock interactive visuals

Exam focus: Uncorrelated is weaker than independent except for the joint normal. Use the Jacobian formula for transformations and the standard order-statistic densities.

Practice

Part C2025 Juneshow ▾
Let X and Y be independent and identically distributed N(0, 1) random variables. Let S=X2+Y2S = X^{2} + Y^{2} and T = e−(X2+Y2)/2e^{-(X^{2}+Y^{2})/2}. Then, which of the following statements are true?
  1. A.The probability density function of S is f_S(s) = (1/2)e−s/2(1/2)e^{-s/2} if s > 0, and 0 otherwise.✓ Correct
  2. B.The probability density function of T is f_T(t) = 1 if 0 < t < 1, and 0 otherwise.✓ Correct
  3. C.Var(S) = 2.
  4. D.E(T) = 2/3.

The worked solution and the per-option analysis are in Trap Analysis.

Part B2025 Decembershow ▾
Let X, Y, and Z be independent Normal random variables with means −1, 0, and 1, respectively, and variances 1, 1, and 3, respectively. Which of the following random variables has a Cauchy distribution with location parameter 0 and scale parameter 1?
  1. A.(X−1)/|Y|
  2. B.(X+2Y+Z)/(X−2Y+Z)✓ Correct
  3. C.(Z−1)/Y
  4. D.(X−Y)/(X+Y)

The worked solution and the per-option analysis are in Trap Analysis.

Part B2025 Juneshow ▾
Let X be a random sample of size 1 from the probability density function f(x|θ)=(3/θ3)(θ−x)2\theta) = (3/\theta^{3})(\theta - x)^{2} if 0<x<θ0 < x < \theta, and 0 otherwise. If (X/(1−λ1),X/(1−λ2))(X/(1 - \lambda_{1}), X/(1 - \lambda_{2})) is a confidence interval for θ\theta with confidence coefficient 1−α1 - \alpha, where λi∈(0,1),i=1,2,λ1<λ2\lambda_{i} \in (0, 1), i = 1, 2, \lambda_{1} < \lambda_{2}, and α∈(0,1)\alpha \in (0, 1), then which of the following statements is true?
  1. A.λ22−λ12=1−α\lambda_{2}^{2} - \lambda_{1}^{2} = 1 - \alpha
  2. B.λ23−λ13=1−α\lambda_{2}^{3} - \lambda_{1}^{3} = 1 - \alpha✓ Correct
  3. C.λ22−λ12=4(1−α)\lambda_{2}^{2} - \lambda_{1}^{2} = 4(1 - \alpha)
  4. D.λ23−λ13=9(1−α)\lambda_{2}^{3} - \lambda_{1}^{3} = 9(1 - \alpha)

The worked solution and the per-option analysis are in Trap Analysis.

Part C2025 Juneshow ▾
Let X1,X2,…,XnX_{1}, X_{2}, \dots, X_{n} be independent and identically distributed random variables with the probability density function f(x|θ)=\theta) = e−(x−θ)e^{-(x-\theta)} if x≥θx \ge \theta, and 0 otherwise, where θ∈R\theta \in \mathbb{R} is the unknown parameter. Further, let Y=min⁡(X1,X2,…,Xn)Y = \min(X_{1}, X_{2}, \dots, X_{n}). Which of the following are confidence intervals for θ\theta with the confidence coefficient (1−α)(1 - \alpha), where α∈(0,1)\alpha \in (0, 1)?
  1. A.(Y+(1/n)ln⁡(1−α/2),Y−(1/n)ln⁡(1−α/2))(Y + (1/n)\ln(1 - \alpha/2), Y - (1/n)\ln(1 - \alpha/2))
  2. B.(Y+(1/2n)ln⁡(α),Y)(Y + (1/2n)\ln(\alpha), Y)
  3. C.(Y+(2/n)ln⁡(α/2),Y+(2/n)ln⁡(1−α/2))(Y + (2/n)\ln(\alpha/2), Y + (2/n)\ln(1 - \alpha/2))✓ Correct
  4. D.(Y+(1/n)ln⁡(α/2),Y+(1/n)ln⁡(1−α/2))(Y + (1/n)\ln(\alpha/2), Y + (1/n)\ln(1 - \alpha/2))✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Juneshow ▾
Let X1,X2,X3X_{1}, X_{2}, X_{3} be a random sample from a continuous distribution having cumulative distribution function F(t), probability density function f(t), and failure rate function r(t) = f(t)/(1 − F(t)), t > 0, where F(0) = 0. If r(t) = 1 for all t > 0, then which of the following statements are true?
  1. A.P(max{X1,X2X_{1}, X_{2}} < 1) = 1/(2e)
  2. B.P(min{X1,X2X_{1}, X_{2}} > 1) = 1/(2e)
  3. C.P(min{X1,X2X_{1}, X_{2}} <X3)=2/3< X_{3}) = 2/3✓ Correct
  4. D.P(max{X1,X2X_{1}, X_{2}} <X3)=1/3< X_{3}) = 1/3✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Decembershow ▾
Let X1,X2,…,Xn(n≥3)X_{1}, X_{2}, \dots, X_{n} (n \ge 3) be a random sample from a population with absolutely continuous cumulative distribution function F(·). The corresponding order statistics are X(1:n) < X(2:n) < ⋯ < X(r:n) < ⋯ < X(n:n). For r = 2, 3, define Y(r,n) = nF(X(r:n)). Suppose that Y(r,n) converges in distribution to a random variable Y_r as n→∞,r=2,3n \to \infty, r = 2, 3. Then, which of the following statements are true?
  1. A.Y2Y_{2} follows gamma distribution with E(Y2)=2E(Y_{2}) = 2.✓ Correct
  2. B.E(Y(3,n)) → 3 as n→∞n \to \infty✓ Correct
  3. C.Y3Y_{3} follows beta distribution with E(Y3)=1/3E(Y_{3}) = 1/3.
  4. D.Y(2,n) follows beta distribution with parameters 2 and n − 1.

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Juneshow ▾
Let X and Y be jointly distributed continuous random variables with joint probability density function f(x, y) = x/y if 0 < x < y < 2, and 0 otherwise. Which of the following statements are true?
  1. A.P(X < 1/2 | Y = 1) = 1/4✓ Correct
  2. B.E(Y) = 1/4
  3. C.P(X < Y/2) = 1/4✓ Correct
  4. D.E(Y/X) = 1/4

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Decembershow ▾
Let X1X_{1} and X2X_{2} be random variables having absolutely continuous distribution functions. Let hi(t)h_{i}(t) denote the hazard function of Xi,i=1,2X_{i}, i = 1, 2. If h1(t)≤h2(t)h_{1}(t) \le h_{2}(t) for all t∈Rt \in \mathbb{R}, then which of the following statements are true?
  1. A.P(X1>1)≥P(X2>1)P(X_{1} > 1) \ge P(X_{2} > 1)✓ Correct
  2. B.P(X1>1)≤P(X2>1/2)P(X_{1} > 1) \le P(X_{2} > 1/2)
  3. C.E(X1)≥E(X2)E(X_{1}) \ge E(X_{2}) provided both the expectations exist.✓ Correct
  4. D.h(t)=h1(t)+h2(t),t∈Rh(t) = h_{1}(t) + h_{2}(t), t \in \mathbb{R}, is the hazard function of the random variable Y = min{X1,X2X_{1}, X_{2}}.

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Juneshow ▾
Let X1,X2X_{1}, X_{2} denote lifetimes (in years) of 2 components of an electronic system. Let Y1=X1+X2,Y2=max⁡Y_{1} = X_{1} + X_{2}, Y_{2} = \max{X1,X2X_{1}, X_{2}} and Y3=min⁡Y_{3} = \min{X1,X2X_{1}, X_{2}}. Assume that X1X_{1} and X2X_{2} are independent, each following exponential distribution with probability density function f(x) = (1/2)e^(−x/2) if x > 0, and 0 otherwise. Which of the following statements are true?
  1. A.P(Y1>2)=2eP(Y_{1} > 2) = 2e⁻1^{1}✓ Correct
  2. B.P(Y2>2)=eP(Y_{2} > 2) = e⁻2^{2}
  3. C.P(Y3>2)=eP(Y_{3} > 2) = e⁻2^{2}✓ Correct
  4. D.Var(Y1+Y2+Y3)=32Var(Y_{1} + Y_{2} + Y_{3}) = 32✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Decembershow ▾
Let X1,X2,…,Xn(n≥3)X_{1}, X_{2}, \dots, X_{n} (n \ge 3) be a random sample from a population with absolutely continuous cumulative distribution function F(·). The corresponding order statistics are X(1:n) < X(2:n) < ⋯ < X(r:n) < ⋯ < X(n:n). Define, for r = 2, 3, Y(r,n) = nF(X(r:n)). Suppose that Y(r,n) converges in distribution to a random variable Y_r as n→∞,r=2,3n \to \infty, r = 2, 3. Then, which of the following statements are true?
  1. A.Y3Y_{3} follows gamma distribution with E(Y3)=3E(Y_{3}) = 3.✓ Correct
  2. B.E(Y(2,n)) → 2 as n→∞n \to \infty✓ Correct
  3. C.Y2Y_{2} follows beta distribution with E(Y2)=1/2E(Y_{2}) = 1/2.
  4. D.Y(3,n) follows beta distribution with parameters 3 and n − 1.

The worked solution and the per-option analysis are in Trap Analysis.

Part B2024 Juneshow ▾
Let a point P be chosen at random on the line segment AB of length α\alpha. Let Z1Z_{1} and Z2Z_{2} denote the lengths of line segments AP and BP respectively. Then the value of E(|Z1−Z2Z_{1} - Z_{2}|) is
  1. A.α\alpha
  2. B.2α2\alpha
  3. C.α/2\alpha/2✓ Correct
  4. D.2α/32\alpha/3

The worked solution and the per-option analysis are in Trap Analysis.

Part C2023 Decembershow ▾
Let Xi(i=1,2)X_{i} (i = 1, 2) be independent with densities fi(x)=if_{i}(x) = i e−ixe^{-ix} for x ≥ 0. Consider a series system of two independent components with lifetimes X1X_{1} and X2X_{2}, and let X denote the lifetime of the system. Which of the following statements are true?
  1. A.P(X > 4) = P(X > 1)P(X > 2)
  2. B.P(X > 4 | X > 2) = P(X > 2)✓ Correct
  3. C.E(X) = 1/3✓ Correct
  4. D.6X ~ χ32\chi^{2}_{3}

The worked solution and the per-option analysis are in Trap Analysis.

Part Bshow ▾
Let X and Y be independent N(0, 1) random variables. The distribution of X + Y is
  1. A.N(0, 1)
  2. B.N(0, 2)✓ Correct
  3. C.N(0, 4)
  4. D.not normal

Solution

A sum of independent normals is normal, with mean 0 + 0 = 0 and variance 1 + 1 = 2 — variances add, standard deviations do not: SD(X+Y)=2(X + Y) = \sqrt{2}, not 2.

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Part Cshow ▾
Let U1,…,U5U_{1}, \dots, U_{5} be i.i.d. Uniform(0, 1) with order statistics U(1)≤⋯≤U(5)U_{(1)} \le \cdots \le U_{(5)}. Which of the following are true?
  1. A.P(U(5)≤x)=x5P(U_{(5)} \le x) = x^{5} for x ∈ [0, 1].✓ Correct
  2. B.U(1)U_{(1)} has density 5(1−x)45(1 - x)^{4} on [0, 1].✓ Correct
  3. C.E[U(k)]=k/6E[U_{(k)}] = k/6 for k = 1, …, 5.✓ Correct
  4. D.U(1)U_{(1)} and U(5)U_{(5)} are independent.

The worked solution and the per-option analysis are in Trap Analysis.