Counterexample bank for CSIR NET Maths Part C

Counterexample bank

Part C is won by knowing which tempting claims are false. 148 counterexamples; 85 free. The rest come with the Notes pack.

#1 · Probability & Statistics › Limit Theorems and Markov Chains

“Convergence in probability ⇒ almost sure convergence” — false

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probabilityconvergence

#2 · Probability & Statistics › Probability

“Uncorrelated ⇒ independent” — false

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probability

#3 · Probability & Statistics › Probability

“Pairwise independent events are mutually independent” — false

Counterexample: Two fair coin tosses: A = first is heads, B = second is heads, C = the two agree

Each pair is independent, but P(A∩B∩C) = 1/4 ≠ 1/8 = P(A)P(B)P(C).

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#4 · Probability & Statistics › Probability

“Every random variable has a moment generating function” — false

Counterexample: The standard Cauchy distribution

E[ $e^{tX}] = \infty$ for every t ≠ 0; even E|X| is infinite. Its characteristic function $e^{- ∣ t ∣}$ exists.

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#5 · Probability & Statistics › Probability

“A mixture of two distributions is a linear combination of the variables” — false

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probability

#6 · Probability & Statistics › Probability

“The maximum of independent exponentials is exponential” — false

Counterexample: max(X₁, X₂) with Xᵢ ~ Exp(1)

The minimum is exponential (rate $λ_{1} + λ_{2})$ ; the maximum has CDF (1 − $e^{- x})^{2}$ , which is not exponential.

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#7 · Probability & Statistics › Probability

“If X and Y are each normal and uncorrelated then they are independent” — false

Counterexample: X ~ N(0,1), ε = ±1 with probability ½ independent of X, Y = εX

Y is N(0,1), Cov(X,Y) = 0, but |X| = |Y| always. The pair is not jointly normal.

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#8 · Probability & Statistics › Probability

“Uncorrelated implies independent” — false

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probability

#9 · Probability & Statistics › Limit Theorems and Markov Chains

“Convergence in probability implies almost sure convergence” — false

Counterexample: The typewriter sequence on [0,1]

$P (X_{n} \neq = 0) \to 0$ but every $ω$ is hit infinitely often, so there is no a.s. limit.

probabilityconvergence

#10 · Probability & Statistics › Limit Theorems and Markov Chains

“Xₙ → 0 almost surely implies E[Xₙ] → 0” — false

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probabilityconvergence

#11 · Probability & Statistics › Limit Theorems and Markov Chains

“An irreducible chain with a stationary distribution converges to it” — false

Counterexample: The two-state chain that swaps deterministically (period 2)

$π = ($ ½, ½) is stationary and unique, but $p_{nij}$ oscillates between 0 and 1. Aperiodicity is required.

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#12 · Probability & Statistics › Limit Theorems and Markov Chains

“A recurrent chain has a stationary distribution” — false

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probabilitymarkov