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Probability, Statistics & Linear Programming

Linear Models and Multivariate

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1. Gauss–Markov, regression, ANOVA basics

Read this as a chapter →

Simple Linear Regression (Part A)

NPTEL · Regression Analysis

OLS and the Gauss–Markov assumptions.

Multiple Linear Regression (Part A)

NPTEL · Regression Analysis

Linear models, Gauss–Markov and ANOVA free

In plain words

Least squares is best among estimators that are linear in the data and unbiased, with no normality assumed. Normality is what buys the tt and FF tests, which is a separate claim.

Gauss–Markov

For Y=Xβ+εY = X\beta + \varepsilon with E[ε]=0,Cov(ε)=σ2IE[\varepsilon] = 0, Cov(\varepsilon) = \sigma^{2}I and X of full rank, the OLS estimator β\betâ = (XᵀX)⁻1XTY^{1}X^{T}Y is the BLUE. Normality is not needed for this — only for the t and F distributions.

Heteroscedasticity: if Var(εi)=σ2xi2Var(\varepsilon_{i}) = \sigma^{2}x_{i}^{2}, divide through by xix_{i}; the BLUE of β\beta in Yi=βxi+εiY_{i} = \beta{}x_{i} + \varepsilon_{i} becomes (1/n)∑(Yi/xi)(1/n)\sum(Y_{i}/x_{i}), not the OLS formula.

Distribution theory (normal errors)

−β- \betâ ~ N(β,σ2(XTX)N(\beta, \sigma^{2}(X^{T}X)⁻1)^{1}).

  • RSS/σ2=(Y−Xβ/\sigma^{2} = (Y - X\betâ)T(Y−Xβ)^{T}(Y - X\betâ)/σ2)/\sigma^{2} ~ **χn−p−12∗∗\chi^{2}_{n-p-1}**, independent of β\betâ (and of XβX\betâ).
  • Total and regression sums of squares are non-central χ2\chi^{2} unless the corresponding parameters vanish.

R2R^{2} and adjusted R2R^{2}

R2=R^{2} = SSR/SST; adjusted R2=1−(1−R2)(n−1)/(n−p−1)R^{2} = 1 - (1 - R^{2})(n-1)/(n-p-1). Adjusted R2R^{2} can decrease when a useless predictor is added — that is its point.

ANOVA and estimability

One-way model Yij=μ+αi+εijY_{ij} = \mu + \alpha_{i} + \varepsilon_{ij} is overparametrised: only functions of the cell means μ+αi\mu + \alpha_{i} are estimable.

Function Estimable?
μ+α1\mu + \alpha_{1} ✓
α1−α2(\alpha_{1} - \alpha_{2} (any contrast) ✓
2μ+α1+α22\mu + \alpha_{1} + \alpha_{2} ✓
μ\mu alone ✗
α1\alpha_{1} alone ✗
μ+α1+α2\mu + \alpha_{1} + \alpha_{2} ✗

A linear function cTβc^{T}\beta is estimable ⇔ c is in the row space of X.

Design degrees of freedom

Design Error d.f.
CRD, k treatments, N units N − k
RBD, k treatments, r blocks (k−1)(r−1)
LSD, p × p (p−1)(p−2)

Key takeaways

  • Gauss–Markov needs equal variances; otherwise weight by 1/variance.
  • RSS/σ2/\sigma^{2} is central χ2\chi^{2} with n − p − 1 d.f.; the other sums of squares are generally non-central.
  • In ANOVA only contrasts and cell means are estimable.

What Gauss–Markov assumes, and what it does notinteractive

This visual is part of Trap Analysis

BLUE without any normality — and what normality actually buys you.

Unlock interactive visuals

Exam focus: Gauss–Markov needs uncorrelated errors of equal variance; with heteroscedasticity the BLUE is weighted least squares. Estimability in ANOVA is decided by whether the function is a combination of cell means.

Practice

Part C2025 Juneshow ▾
Consider a paired data (xi,yi);i=1,2,3,4,5(x_{i}, y_{i}); i = 1, 2, 3, 4, 5, where (x1,x2,x3,x4,x5)=(−2,−1,0,1,2)(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}) = (-2, -1, 0, 1, 2) and yi=xi2y_{i} = x_{i}^{2} for all i = 1, 2, 3, 4, 5. On this data, a simple linear regression model with an intercept term and a simple linear regression model without an intercept term are fitted using the method of least squares. Which of the following statements are true?
  1. A.The two fitted lines have the same slope✓ Correct
  2. B.The two fitted models have the same intercept
  3. C.The model with intercept passes through at least one of the observed data points
  4. D.The model without intercept passes through at lesast one of the observed data points✓ Correct

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

Part B2025 Juneshow ▾
Consider the multiple linear regression model yi=β0+β1x1i+⋯+β8x8i+εi,i=1,2,…,29y_{i} = \beta_{0} + \beta_{1}x_{1i} + \dots + \beta_{8}x_{8i} + \varepsilon_{i}, i = 1, 2, \dots, 29, where ε1,ε2,…,ε29\varepsilon_{1}, \varepsilon_{2}, \dots, \varepsilon_{29} are independent and identically normal distributed with mean 0 and variance σ2\sigma^{2}. Suppose the model is fitted using the method of least squares. If the calculated value of the F-statistic for testing the significance of regression is 2.50, then the possible values of R2R^{2} and Adjusted R2R^{2} are respectively
  1. A.0.30 and 0.10
  2. B.0.50 and 0.30✓ Correct
  3. C.0.50 and 0.16
  4. D.0.30 and −0.10

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

Part B2025 Decembershow ▾
Consider the linear model Y1=β1+β2+ε1,Y2=β1+2β2+ε2,Y3=β1+cβ2+ε3Y_{1} = \beta_{1}+\beta_{2}+\varepsilon_{1}, Y_{2} = \beta_{1}+2\beta_{2}+\varepsilon_{2}, Y_{3} = \beta_{1}+c\beta_{2}+\varepsilon_{3}, where β1,β2∈R\beta_{1},\beta_{2} \in \mathbb{R} are unknown parameters, and the uncorrelated errors εi,i=1,2,3\varepsilon_{i}, i=1,2,3 have zero mean and finite variance σ2(>0)\sigma^{2}(>0). The constant c is such that β\betâ1_{1} and β\betâ2_{2} are uncorrelated, where β\betâ1_{1} and β\betâ2_{2} are the best linear unbiased estimators of β1\beta_{1} and β2\beta_{2}, respectively. Which of the following statements is the correct option for (Var(β(Var(\betâ1),Var(β_{1}), Var(\betâ2))_{2}))?
  1. A.(σ2/3,σ2/14)(\sigma^{2}/3, \sigma^{2}/14)✓ Correct
  2. B.(3σ2,14σ2)(3\sigma^{2}, 14\sigma^{2})
  3. C.(σ2/14,σ2/3)(\sigma^{2}/14, \sigma^{2}/3)
  4. D.(14σ2,3σ2)(14\sigma^{2}, 3\sigma^{2})

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

Part C2025 Decembershow ▾
Consider the simple linear regression model Yi=βxi+εi,i=1,2,…,nY_{i}=\beta{}x_{i}+\varepsilon_{i}, i=1,2,\dots,n, where β>0,∑xi2>0\beta>0, \sum{}x_{i}^{2}>0, and the uncorrelated errors εi\varepsilon_{i} have zero mean and finite variance σ2(>0)\sigma^{2}(>0). Let β\betã1=∑ai_{1}=\sum{}a_{i}YiY_{i}, where aia_{i}'s minimize E(∑aiYi−β)2E(\sum{}a_{i} Y_{i}-\beta)^{2} with respect to scalars a1,a2,…,ana_{1},a_{2},\dots,a_{n}. Let β\betã2_{2} be the ordinary least squares estimator of β\beta. Which of the following statements are true?
  1. A.Var(βVar(\betã1)>Var(β_{1}) > Var(\betã2),E(β_{2}), E(\betã1−β)2>E(β_{1}-\beta)^{2} > E(\betã2−β)2_{2}-\beta)^{2}
  2. B.Var(βVar(\betã1)<Var(β_{1}) < Var(\betã2),E(β_{2}), E(\betã1−β)2<E(β_{1}-\beta)^{2} < E(\betã2−β)2_{2}-\beta)^{2}✓ Correct
  3. C.Var(βVar(\betã1)>Var(β_{1}) > Var(\betã2),E(β_{2}), E(\betã1)<E(β_{1}) < E(\betã2)_{2})
  4. D.Var(βVar(\betã1)<Var(β_{1}) < Var(\betã2),E(β_{2}), E(\betã1)>E(β_{1}) > E(\betã2)_{2})

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

Part C2025 Decembershow ▾
Let Y=Xβ+εY=X\beta+\varepsilon be a multiple linear regression model with p regressors and an intercept, where β=(β0,β1,…,βp)T\beta=(\beta_{0},\beta_{1},\dots,\beta_{p})^{T} and the random error ε\varepsilon~Nn(0,σ2I),σ>0N_{n}(0,\sigma^{2}I), \sigma>0 and n>p+1. The least squares method provides a unique estimator β\betâ. Let the total sum of squares (corrected), sum of squares due to regression, and sum of squares due to error, based on β\betâ, be denoted YᵀAY, YᵀBY and YᵀCY respectively, so YᵀAY=YᵀBY+YᵀCY. Which of the following statements are always true?
  1. A.YᵀAY/σ2/\sigma^{2} follows a central χ2\chi^{2} distribution with (n−1) degrees of freedom.
  2. B.YᵀBY/σ2/\sigma^{2} follows a central χ2\chi^{2} distribution with p degrees of freedom if β1=β2=⋯=βp=0\beta_{1}=\beta_{2}=\cdots=\beta_{p}=0.✓ Correct
  3. C.YᵀBY/YᵀCY follows a central F distribution with (p,n−p) degrees of freedom.
  4. D.YᵀBY and YᵀCY are independently distributed if and only if β1=β2=⋯=βp=0\beta_{1}=\beta_{2}=\cdots=\beta_{p}=0.

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

Part C2025 Juneshow ▾
An analyst fits a multiple linear regression model Yi=β0+β1X1i+⋯+β4X4i+εi,i=1,2,…,nY_{i} = \beta_{0} + \beta_{1}X_{1i} + \dots + \beta_{4}X_{4i} + \varepsilon_{i}, i = 1, 2, \dots, n, using the method of least squares. However, his coordinator insists that the intercept and two regressors Z1=X1+X3Z_{1} = X_{1} + X_{3} and Z2=X2−X4Z_{2} = X_{2} - X_{4} are enough to represent the model. Suppose the coordinator's claim can be tested in the form of a general linear hypothesis, viz., H0:LTβ=0H_{0} : L^{T}\beta = 0 against HA:H0H_A : H_{0} is not true, where βT=(β0,β1,…,β4)\beta^{T} = (\beta_{0}, \beta_{1}, \dots, \beta_{4}). Suppose we have n observations on the response Y and each regressor. Further, assume that the errors with or without restrictions are independent N(0,σ2)N(0, \sigma^{2}) variables. Then, which of the following statements are true?
  1. A.A possible choice of L is the 5 × 2 matrix with rows (0, 0), (1, 0), (0, 1), (−1, 0), (0, 1)✓ Correct
  2. B.The test statistic for testing H0H_{0} against H_A follows an F-distribution with (2, n − 4) degrees of freedom under H0H_{0}
  3. C.Sum of squares residuals under the restrictions LTβ=0L^{T}\beta = 0 follows σ2χn−32\sigma^{2}\chi^{2}_{n-3} distribution✓ Correct
  4. D.Sum of squares residuals without the restrictions LTβ=0L^{T}\beta = 0 follows a non-central σ2χn−42\sigma^{2}\chi^{2}_{n-4} distribution

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

Part B2024 Decembershow ▾
Consider a multiple linear regression model Yi=β0+β1xi1+⋯+βpxip+εi,1≤i≤n,n>(p+1)Y_{i} = \beta_{0} + \beta_{1}x_{i1} + \cdots + \beta_p x_ip + \varepsilon_{i}, 1 \le i \le n, n > (p + 1), where the errors εi′s\varepsilon_{i}'s are uncorrelated with zero mean and finite variance σ2>0\sigma^{2} > 0. Here, YiY_{i} is the i-th response. Let Ŷi_{i} be the i-th predicted response by the least squares estimation method, and let ε\varepsilon̂i=Yi−_{i} = Y_{i} - Ŷi,1≤i≤n_{i}, 1 \le i \le n. Then, which of the following statements is true?
  1. A.Var(εVar(\varepsilon̂i)≤Var(εi),1≤i≤n_{i}) \le Var(\varepsilon_{i}), 1 \le i \le n✓ Correct
  2. B.Cov(εCov(\varepsilon̂i,ε_{i}, \varepsilon̂k)=Cov(εi,εk)_{k}) = Cov(\varepsilon_{i}, \varepsilon_{k}), for all i ≠ k = 1, 2, …, n
  3. C.Var(Ŷi)=Var(Yi),1≤i≤n_{i}) = Var(Y_{i}), 1 \le i \le n
  4. D.E(Ŷi)<E(Yi),1≤i≤n_{i}) < E(Y_{i}), 1 \le i \le n

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

Part C2024 Juneshow ▾
In a standard linear regression model, let R2R^{2} and Rˉ2{\bar{R}}^{2}, respectively, denote the coefficient of determination and adjusted coefficient of determination. Which of the following statements are true?
  1. A.Rˉ2<R2{\bar{R}}^{2} < R^{2}✓ Correct
  2. B.R2R^{2} increases as the number of independent variables increase✓ Correct
  3. C.Rˉ2{\bar{R}}^{2} decreases as the number of independent variables increase
  4. D.Rˉ2>0{\bar{R}}^{2} > 0

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

Part B2024 Juneshow ▾
An analyst considers standardized values of observations on three variables, consumption (C), saving (S) and total income (TI) so that they have zero means and unit variances. She further considers disposable income (DI) where DI = C + S. In the simple linear regressions of DI on TI, DI on C and S on TI, the regression coefficients are 0.8, 0.5 and 0.4, respectively. There are 21 sample observations. Sample covariances and variances are calculated with divisor 20. Then, the value of sum of squared residuals in the regression of DI on S is
  1. A.5
  2. B.10
  3. C.15✓ Correct
  4. D.20

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

Part B2024 Decembershow ▾
Consider a multiple linear regression model Yi=β0+β1xi1+⋯+βpxip+εi,1≤i≤n,n>(p+1)Y_{i} = \beta_{0} + \beta_{1}x_{i1} + \cdots + \beta_p x_ip + \varepsilon_{i}, 1 \le i \le n, n > (p + 1), where errors εi′s\varepsilon_{i}'s are uncorrelated with zero mean and finite variance σ2>0\sigma^{2} > 0. Here, YiY_{i} is the i-th response. Let Ŷi_{i} be the i-th predicted response by the least squares estimation method, and let ε\varepsilon̂i=Yi−_{i} = Y_{i} - Ŷi,1≤i≤n_{i}, 1 \le i \le n. Then, which of the following statements is true?
  1. A.Var(Ŷi)≤Var(Yi),1≤i≤n_{i}) \le Var(Y_{i}), 1 \le i \le n✓ Correct
  2. B.Cov(Ŷi_{i}, Ŷk)=Cov(Yi,Yk),1≤i<k≤n_{k}) = Cov(Y_{i}, Y_{k}), 1 \le i < k \le n
  3. C.Var(εVar(\varepsilon̂i)=Var(εi),1≤i≤n_{i}) = Var(\varepsilon_{i}), 1 \le i \le n
  4. D.E(εE(\varepsilon̂i)<E(εi),1≤i≤n_{i}) < E(\varepsilon_{i}), 1 \le i \le n

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

Part C2024 Decembershow ▾
Consider a linear model Yi=β1+β2+⋯+βi+εi,1≤i≤nY_{i} = \beta_{1} + \beta_{2} + \cdots + \beta_{i} + \varepsilon_{i}, 1 \le i \le n, where errors εi′s\varepsilon_{i}'s are uncorrelated with zero mean and finite variance σ2>0\sigma^{2} > 0. Let β\betâi_{i} be the best linear unbiased estimator (BLUE) of βi,i=1,2,…,n\beta_{i}, i = 1, 2, \dots, n. Then, which of the following statements are true?
  1. A.The sum of squares residuals is strictly positive with probability 1.
  2. B.For every βi,1≤i≤n\beta_{i}, 1 \le i \le n, there are infinitely many linear unbiased estimators.
  3. C.Var(βVar(\betâ1)=σ2_{1}) = \sigma^{2}✓ Correct
  4. D.Y3−Y2Y_{3} - Y_{2} is the BLUE of β3\beta_{3}.✓ Correct

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

Part C2024 Decembershow ▾
Consider a linear regression model Y=Xβ+εY = X\beta + \varepsilon, with r regressors and an intercept. Random error ε\varepsilon ~ Nn(0,σ2In)N_{n}(0, \sigma^{2}I_{n}) and X has full column rank. Here InI_{n} denotes the identity matrix of order n. Regression coefficients are estimated by the least squares estimation method. Let σ\sigmâ2^{2} and σ\sigmâ2(^{2}(MLE), respectively, be the mean squares residuals and the maximum likelihood estimator of σ2\sigma^{2}. Then, which of the following statements are true?
  1. A.MSE(σ(\sigmâ2(^{2}(MLE)) < MSE(σ(\sigmâ2)^{2}) if r = 3, n = 12✓ Correct
  2. B.Var(σVar(\sigmâ2(^{2}(MLE))<Var(σ)) < Var(\sigmâ2)^{2}) if 1 ≤ r ≤ n − 2, n ≥ 3✓ Correct
  3. C.Var(σVar(\sigmâ2(^{2}(MLE))>Var(σ)) > Var(\sigmâ2)^{2}) if 1 ≤ r ≤ 6, n ≥ 12
  4. D.MSE(σ(\sigmâ2(^{2}(MLE)) > MSE(σ(\sigmâ2)^{2}) if r = 6, n = 12✓ Correct

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

Part C2024 Decembershow ▾
Consider a linear model Yi=β1+β2+⋯+βi+εi,1≤i≤nY_{i} = \beta_{1} + \beta_{2} + \cdots + \beta_{i} + \varepsilon_{i}, 1 \le i \le n, where the errors εi′s\varepsilon_{i}'s are uncorrelated with zero mean and finite variance σ2>0\sigma^{2} > 0. Then, which of the following statements are true?
  1. A.Every linear function of βi,1≤i≤n\beta_{i}, 1 \le i \le n, is not estimable.
  2. B.Every βi,1≤i≤n\beta_{i}, 1 \le i \le n, has infinitely many linear unbiased estimators, but a unique best linear unbiased estimator.
  3. C.Each βi,1≤i≤n\beta_{i}, 1 \le i \le n, has only one linear unbiased estimator.✓ Correct
  4. D.Y2−Y1Y_{2} - Y_{1} is the best linear unbiased estimator of β2\beta_{2}.✓ Correct

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

Part C2024 Juneshow ▾
Consider the two-way ANOVA model Yij=μ+αi+βj+εij,i=1,2;j=1,2Y_{ij} = \mu + \alpha_{i} + \beta_{j} + \varepsilon_{ij}, i = 1, 2; j = 1, 2, where μ\mu is the overall mean effect, αi\alpha_{i} is the effect of the i-th level of factor A,βjA, \beta_{j} is the effect of j-th level of factor B,YijB, Y_{ij} is the response of the (i, j)-th experimental unit and εij\varepsilon_{ij} is the corresponding error with E(εij)=0E(\varepsilon_{ij}) = 0 for i = 1, 2; j = 1, 2. Which of the following are estimable linear parametric functions?
  1. A.μ+α2+β2\mu + \alpha_{2} + \beta_{2}✓ Correct
  2. B.α1−β1\alpha_{1} - \beta_{1}
  3. C.α2−β2\alpha_{2} - \beta_{2}
  4. D.μ−α1−β1\mu - \alpha_{1} - \beta_{1}

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

Part C2024 Decembershow ▾
Consider a linear regression model Y=Xβ+εY = X\beta + \varepsilon, with r regressors and an intercept. Random error ε\varepsilon ~ Nn(0,σ2In)N_{n}(0, \sigma^{2}I_{n}) and X has full column rank. Here InI_{n} denotes the identity matrix of order n. Regression coefficients are estimated by the least squares estimation method. Let σ\sigmâ2^{2} and σ\sigmâ2(^{2}(MLE), respectively, be the mean squares residuals and the maximum likelihood estimator of σ2\sigma^{2}. Then, which of the following statements are true?
  1. A.MSE(σ(\sigmâ2(^{2}(MLE)) < MSE(σ(\sigmâ2)^{2}) if r = 2, n = 12✓ Correct
  2. B.Var(σVar(\sigmâ2(^{2}(MLE))>Var(σ)) > Var(\sigmâ2)^{2}) if 2 ≤ r ≤ n − 2, n ≥ 12
  3. C.Var(σVar(\sigmâ2(^{2}(MLE))<Var(σ)) < Var(\sigmâ2)^{2}) if 1 ≤ r ≤ n − 2, n ≥ 3✓ Correct
  4. D.MSE(σ(\sigmâ2(^{2}(MLE)) > MSE(σ(\sigmâ2)^{2}) if r = 7, n = 12

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

Part C2023 Decembershow ▾
Consider the one-way fixed effects ANOVA model Yij=μ+αi+εij,j=1,…,ni;i=1,…,kY_{ij} = \mu + \alpha_{i} + \varepsilon_{ij}, j = 1, \dots, n_{i}; i = 1, \dots, k, with uncorrelated errors of mean 0 and variance σ2\sigma^{2}. Let Ȳi_{i} be the i-th group mean. Which of the following statements are true?
  1. A.(1/∑ni)∑i∑jYij(1/\sum{}n_{i}) \sum_{i}\sum_{j} Y_{ij} is an unbiased estimator of μ\mu
  2. B.2μ+α1+α22\mu + \alpha_{1} + \alpha_{2} is an estimable linear parametric function✓ Correct
  3. C.μ+α1+α2\mu + \alpha_{1} + \alpha_{2} is an estimable linear parametric function
  4. D.(1/n2)∑j(Y2j−(1/n_{2}) \sum_{j} (Y_{2j} - Ȳ2)_{2}) is an unbiased estimator of α2\alpha_{2}

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

Part B2023 Decembershow ▾
Consider Y=α+βx+εY = \alpha + \beta{}x + \varepsilon with E(ε)=0E(\varepsilon) = 0. From 10 observations the OLS fit is ŷ = 1.5 + 0.8x. Suppose ∑(yi−\sum(y_{i} - ȳ)2=5)^{2} = 5 and ∑(xi−xˉ)2=6\sum(x_{i} - {\bar{x}})^{2} = 6. The adjusted R2(R^{2} (to two decimals) equals
  1. A.0.74✓ Correct
  2. B.0.83
  3. C.0.77
  4. D.0.84

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

Part B2023 Juneshow ▾
Consider the simple linear regression model Yi=βxi+εi,i=1,…,nY_{i} = \beta{}x_{i} + \varepsilon_{i}, i = 1, \dots, n, where E(εi)=0,Cov(εi,εk)=0E(\varepsilon_{i}) = 0, Cov(\varepsilon_{i}, \varepsilon_{k}) = 0 for i ≠ k and Var(εi)=xi2σ2Var(\varepsilon_{i}) = x_{i}^{2}\sigma^{2}. The best linear unbiased estimator of β\beta is
  1. A.∑Yixi/∑xi2\sum Y_{i}x_{i} / \sum x_{i}^{2}
  2. B.∑Yi/∑xi\sum Y_{i} / \sum x_{i}
  3. C.(1/n)∑Yi/xi(1/n) \sum Y_{i}/x_{i}✓ Correct
  4. D.(1/n)∑Yixi/xi2(1/n) \sum Y_{i}x_{i} / x_{i}^{2}

Solution

Heteroscedastic errors: divide by xix_{i} to get Yi/xi=β+εi/xiY_{i}/x_{i} = \beta + \varepsilon_{i}/x_{i} with constant variance σ2\sigma^{2}; the BLUE is the plain mean of Yi/xi(Y_{i}/x_{i} (weighted least squares with weights 1/xi2)1/x_{i}^{2}).

Save for spaced revision (free account)
Part C2023 Decembershow ▾
Consider Y=Xβ+εY = X\beta + \varepsilon with X a fixed n × (p + 1) matrix of rank p + 1 and ε\varepsilon ~ Nn(0,σ2I)N_{n}(0, \sigma^{2}I). If β\betâ is the OLS estimator, which of the following statements are true?
  1. A.(1/σ2)YTXβ(1/\sigma^{2}) Y^{T}X\betâ has a central χp+12\chi^{2}_{p+1} distribution
  2. B.(1/σ2)(Y−Xβ(1/\sigma^{2})(Y - X\betâ)T(Y−Xβ)^{T}(Y - X\betâ) has a central χn−p−12\chi^{2}_{n-p-1} distribution✓ Correct
  3. C.XβX\betâ and (Y−Xβ(Y - X\betâ)T(Y−Xβ)^{T}(Y - X\betâ) are independently distributed✓ Correct
  4. D.(1/σ2)∑(Yi−(1/\sigma^{2}) \sum(Y_{i} - Ȳ)2)^{2} has a central χn−12\chi^{2}_{n-1} distribution

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

Part Cshow ▾
Consider y=Xβ+εy = X\beta + \varepsilon with X of full column rank,Eε=0,Covε=σ2Irank, E \varepsilon = 0, Cov \varepsilon = \sigma^{2}I. Which of the following are true?
  1. A.The OLS estimator is β\betâ = (XᵀX)⁻1XTy^{1}X^{T}y.✓ Correct
  2. B.β\betâ is the best linear unbiased estimator of β\beta.✓ Correct
  3. C.The Gauss–Markov theorem requires ε\varepsilon to be normally distributed.
  4. D.The residual vector y−Xβy - X\betâ is orthogonal to every column of X.✓ Correct

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

2. Multivariate normal distribution

Read this as a chapter →

Lec-03 Multivariate normal distribution - I

NPTEL · Applied Multivariate Analysis

All linear combinations normal ⇔ jointly normal.

Lec-11 Wishart distribution and it's properties - I

NPTEL · Applied Multivariate Analysis

Quadratic forms cᵀAc — get the scaling right.

Lec-13 Hotelling's T2 distribution and it's applications

NPTEL · Applied Multivariate Analysis

Multivariate normal and Wishart free

In plain words

Every linear combination is normal again, and so is every conditional, with a mean linear in whatever you conditioned on. It is the one family where uncorrelated really does mean independent.

Multivariate normal

X ~ Np(μ,∑)N_p(\mu, \sum). Key properties:

  • Every linear combination aᵀX is univariate normal — and this characterises the MVN.
  • AX + b ~ N(Aμ+b,A∑AT)N(A\mu + b, A\sum{}A^{T}).
  • Marginals are normal; conditionals are normal with E[X1E[X_{1}|X2]=μ1+∑12∑22X_{2}] = \mu_{1} + \sum_{12}\sum_{22}⁻1(X2−μ2),Cov=∑11−∑12∑22^{1}(X_{2} - \mu_{2}), Cov = \sum_{11} - \sum_{12}\sum_{22}⁻1∑21^{1}\sum_{21}.
  • Uncorrelated ⇒ independent (only for the joint normal). −(X−μ)T∑- (X - \mu)^{T}\sum⁻1(X−μ)^{1}(X - \mu) ~ χp2\chi^{2}_p.

Quick computation: for X ~ N2(0,[[5,−3],[−3,10]])N_{2}(0, [[5,-3],[-3,10]]) and Y=(X1,5−2X2)Y = (X_{1}, 5 - 2X_{2}): mean (0,5);Var(5−2X2)=4⋅10=40;Cov=−2(−3)=6(0, 5); Var(5 - 2X_{2}) = 4\cdot10 = 40; Cov = -2(-3) = 6.

Wishart

A=∑(Xi−Xˉ)(Xi−Xˉ)TA = \sum(X_{i} - {\bar{X}})(X_{i} - {\bar{X}})^{T} ~ Wp(n−1,∑)W_p(n - 1, \sum) for a sample of size n from Np(μ,∑)N_p(\mu, \sum).

  • Quadratic forms: cᵀAc ~ (cT∑c)⋅χn−12(c^{T}\sum{}c)\cdot\chi^{2}_{n-1} for a fixed vector c. Get the scaling right — this is the most-tested fact.
  • Trace(A∑(A\sum⁻1)^{1}) ~ χ(n−1)p2\chi^{2}_{(n-1)p}.
  • X̄ and A are independent (the multivariate Cochran theorem). −Wp(n,∑)- W_p(n, \sum) with ∑=I\sum = I and p = 1 is just χn2\chi^{2}_{n}.
  • Hotelling′sT2's T^{2} =n(Xˉ−μ)TS= n({\bar{X}} - \mu)^{T}S⁻1(Xˉ−μ);T2^{1}({\bar{X}} - \mu); T^{2} ~ (n−1)p/(n−p) · Fp,n−pF_{p, n-p}.
  • Sample correlation of a bivariate normal: the population Spearman correlation is (6/π)(6/\pi)arcsin(ρ/2)(\rho/2), Kendall′sτ's \tau is (2/π)(2/\pi)arcsin ρ\rho.

Key takeaways

  • All linear combinations normal ⇔ jointly normal. −cTWp(n,∑)c=(cT∑c)χn2- c^{T} W_p(n,\sum) c = (c^{T}\sum{}c)\chi^{2}_{n} — compute cT∑cc^{T}\sum{}c carefully.
  • Uncorrelated ⇒ independent only in the joint normal world.

Zero correlation and independence: when they coincideinteractive

This visual is part of Trap Analysis

The equivalence holds under joint normality and fails without it — with the standard counterexample.

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Exam focus: Linear combinations of a multivariate normal are normal — that single fact plus the Wishart quadratic-form rule answers most questions here.

Practice

Part C2025 Juneshow ▾
Let (X1,Y1),(X2,Y2),…,(X10,Y10)(X_{1}, Y_{1}), (X_{2}, Y_{2}), \dots, (X_{10}, Y_{10}) be a random sample from a bivariate normal distribution BVN(μ1,μ2,σ12,σ22,ρ)(\mu_{1}, \mu_{2}, \sigma_{1}^{2}, \sigma_{2}^{2}, \rho) with μ1=5,μ2=6,σ12=4,σ22=9\mu_{1} = 5, \mu_{2} = 6, \sigma_{1}^{2} = 4, \sigma_{2}^{2} = 9 and ρ=1/2\rho = 1/2. Then, which of the following statements are true?
  1. A.The distribution of (1/7)∑i(1/\sqrt{7})\sum_{i}₌110(Xi−Yi+1)_{1}^{10}(X_{i} - Y_{i} + 1) is N(0, 10)✓ Correct
  2. B.The distribution of (1/19)∑i(1/19)\sum_{i}₌110(Xi+Yi−11)2_{1}^{10}(X_{i} + Y_{i} - 11)^{2} is χ2−\chi^{2}-distribution with degrees of freedom 10✓ Correct
  3. C.The distribution of 22(X1−5)/√(∑i2\sqrt{2}(X_{1} - 5)/\surd(\sum_{i}₌310(Xi−5)2)_{3}^{10}(X_{i} - 5)^{2}) is t-distribution with degrees of freedom 9
  4. D.The distribution of 2∑i2\sum_{i}₌13(Yi−6)2/∑i_{1}^{3}(Y_{i} - 6)^{2}/\sum_{i}₌49(Yi−6)2_{4}^{9}(Y_{i} - 6)^{2} is F-distribution with degrees of freedom 3 and 6.✓ Correct

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

Part C2025 Juneshow ▾
Let the random vector (X1,X2,X3)T(X_{1}, X_{2}, X_{3})^{T} have the positive definite dispersion matrix with rows (1,ρ,ρ),(ρ,1,ρ)(1, \rho, \rho), (\rho, 1, \rho) and (ρ,ρ,1)(\rho, \rho, 1). Then, which of the following statements are true?
  1. A.ρ\rho may be −0.47✓ Correct
  2. B.The first principal component can only explain 32% of the total variation for some ρ\rho
  3. C.The second principal component can explain more than 32% of the total variation for any ρ\rho
  4. D.The variance of the first principal component is 1+2ρ1 + 2\rho for any ρ\rho

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

Part B2025 Juneshow ▾
Suppose (X1,X2,X3)T(X_{1}, X_{2}, X_{3})^{T} ~ N3(0,I3)N_{3}(0, I_{3}), where 0 is the zero mean vector and I3I_{3} is the 3 × 3 identity matrix, and (X,Y,Z)T=A(X1,X2,X3)T(X, Y, Z)^{T} = A(X_{1}, X_{2}, X_{3})^{T} where A has rows (3, 0, 0), (2, 2, 0) and (4, 0, 4). Then the partial correlation coefficient ρYZ.X\rho_{YZ.X} is
  1. A.1/2
  2. B.2/3
  3. C.3/4
  4. D.0✓ Correct

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

Part C2025 Decembershow ▾
Let X1,X2,…,X20X_{1}, X_{2}, \dots, X_{20} be a random sample from N12(μ,∑)N_{12}(\mu,\sum), where μ∈R12\mu\in\mathbb{R}^{12} and ∑\sum is positive definite. Suppose Xˉ=(1/20)∑i{\bar{X}} = (1/20)\sum_{i}₌120Xi_{1}^{20}X_{i} and S=(1/19)∑iS = (1/19)\sum_{i}₌120(Xi−Xˉ)(Xi−Xˉ)T_{1}^{20}(X_{i}-{\bar{X}})(X_{i}-{\bar{X}})^{T}. Then which of the following statements are true?
  1. A.19(X̄ᵀSX)(XˉT∑Xˉ))({\bar{X}}^{T}\sum{\bar{X}})⁻1^{1} ~ χ192\chi^{2}_{19}✓ Correct
  2. B.E(S⁻1)=(19/6)∑^{1}) = (19/6)\sum⁻1^{1}✓ Correct
  3. C.S ~ W12(19,19∑)W_{12}(19, 19\sum)
  4. D.trace(19∑(19\sum⁻1S)^{1}S) ~ χ2282\chi^{2}_{228}✓ Correct

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

Part B2025 Decembershow ▾
Let X=(X1,X2,X3)TX = (X_{1},X_{2},X_{3})^{T} be a 3×1 random vector with E(X) = (3,2,1)ᵀ and Cov(X)=∑Cov(X) = \sum, with rows (3,−2,0), (−2,3,−2), (0,−2,3). Suppose that Y=(Y1,Y2,Y3)T=(1/3)XY = (Y_{1},Y_{2},Y_{3})^{T} = (1/\sqrt{3})X. Then the value of the multiple correlation coefficient between Y1Y_{1} and (Y2,Y3)(Y_{2},Y_{3}) equals
  1. A.2/52/\sqrt{5}✓ Correct
  2. B.2/5\sqrt{2/5}
  3. C.2/32/\sqrt{3}
  4. D.1/31/\sqrt{3}

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

Part B2024 Juneshow ▾
Let X0,X1,…,Xp(p≥2)X_{0}, X_{1}, \dots, X_p (p \ge 2) be independent and identically distributed random variables with mean 0 and variance 1. Suppose Yi=X0+Xi,i=1,…,pY_{i} = X_{0} + X_{i}, i = 1, \dots, p. The first principal component based on the covariance matrix of Y=(Y1,…,Yp)TY = (Y_{1}, \dots, Y_p)^{T} is
  1. A.(1/p)∑i(1/\sqrt{p}) \sum_{i}₌1_{1}ᵖ YiY_{i}✓ Correct
  2. B.(1/p)∑i(1/p) \sum_{i}₌1_{1}ᵖ YiY_{i}
  3. C.p∑i\sqrt{p} \sum_{i}₌1_{1}ᵖ YiY_{i}
  4. D.∑i\sum_{i}₌1_{1}ᵖ YiY_{i}

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

Part C2024 Decembershow ▾
Let (X1,X2)(X_{1}, X_{2}) be a bivariate normal random vector with E(X1)=1,E(X2)=0,Var(X1)=1,Var(X2)=1E(X_{1}) = 1, E(X_{2}) = 0, Var(X_{1}) = 1, Var(X_{2}) = 1, and correlation coefficient 1/2. Let U be a U(0, 1) random variable, which is independent of (X1,X2)(X_{1}, X_{2}). If Z = (UX1+X2−U)/U2+U+1_{1} + X_{2} - U)/\sqrt{U^{2} + U + 1}, then which of the following statements are true?
  1. A.The distribution of Z is symmetric about 0.✓ Correct
  2. B.E(Z2)=2E(Z^{2}) = 2
  3. C.Var(Z2)=1Var(Z^{2}) = 1
  4. D.Z and U are independent random variables.✓ Correct

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

Part C2024 Decembershow ▾
If (X1,X2)(X_{1}, X_{2})ᵗ ~ N2((0,0)N_{2}((0, 0)ᵗ, [[3, 1], [1, 3]]), then which of the following statements are true?
  1. A.The distribution of the second principal component is normal with mean 0 and variance 2.✓ Correct
  2. B.The variance of the first principal component is 4.✓ Correct
  3. C.The first principal component explains more than 70% of the total variance.
  4. D.The correlation coefficient between the first and the second principal components is 0.✓ Correct

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

Part B2024 Decembershow ▾
Let X, Y and Z be random variables such that S = [[X, Y], [Y, Z]] ~ W2(10,∑)W_{2}(10, \sum), where W2W_{2} denotes the Wishart distribution and ∑=[[1,1/2],[1/2,1]]\sum = [[1, 1/2], [1/2, 1]]. Define T=Z−Y2/XT = Z - Y^{2}/X. Then, Var(T) equals
  1. A.77/6
  2. B.81/8✓ Correct
  3. C.83/9
  4. D.79/7

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

Part C2024 Juneshow ▾
Let X=(X1,X2)TX = (X_{1}, X_{2})^{T} be a bivariate random vector with covariance matrix ∑=[[1,2],[2,2]]\sum = [[1, \sqrt{2}], [\sqrt{2}, 2]]. Which of the following statements are true?
  1. A.The first principal component based on ∑\sum explains exactly 90% of the total variability
  2. B.The second principal component based on ∑\sum explains exactly 10% of the total variability
  3. C.sup{aT∑a:a∈R2a^{T}\sum{}a : a \in \mathbb{R}^{2} and aᵀa = 1} = 3✓ Correct
  4. D.The first principal component based on ∑\sum is (1/3)(X1+2X2)(1/\sqrt{3})(X_{1} + \sqrt{2X_{2}})✓ Correct

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

Part B2024 Decembershow ▾
Let X, Y and Z be independent and identically distributed (i.i.d.) random variables with distribution N(0, 1). Define U = 2X, V = 3X + Y, W = X + 4Z. Then the partial correlation coefficient of V and W, given U is
  1. A.0✓ Correct
  2. B.0.5
  3. C.−1
  4. D.1

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

Part C2023 Decembershow ▾
Suppose A=(aij)A = (a_{ij}) ~ W3(5,∑)W_{3}(5, \sum), where ∑=[[2,1,1],[1,2,0],[1,0,2]]\sum = [[2, 1, 1], [1, 2, 0], [1, 0, 2]]. Then which of the following statements are true?
  1. A.a22a_{22} ~ χ32\chi^{2}_{3}
  2. B.½ a22a_{22} ~ χ52\chi^{2}_{5}✓ Correct
  3. C.(1/33)(a11−4a13+4a33)(1/33)(a_{11} - 4a_{13} + 4a_{33}) ~ χ32\chi^{2}_{3}
  4. D.(1/9)(a11−4a13+4a33)(1/9)(a_{11} - 4a_{13} + 4a_{33}) ~ χ52\chi^{2}_{5}

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

Part B2023 Decembershow ▾
For n ≥ p + 1, let X1,…,XnX_{1}, \dots, X_{n} be a random sample from Np(μ,∑)N_{p}(\mu, \sum) with ∑\sum positive definite. Define X̄ and A=∑(Xi−Xˉ)(Xi−Xˉ)TA = \sum(X_{i} - {\bar{X}})(X_{i} - {\bar{X}})^{T}. The distribution of Trace(A∑(A\sum⁻1)^{1}) is
  1. A.Wp(n−1,∑)W_{p}(n - 1, \sum)
  2. B.χp2\chi^{2}_{p}
  3. C.χnp2\chi^{2}_{np}
  4. D.χ(n−1)p2\chi^{2}_{(n-1)p}✓ Correct

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

Part C2023 Decembershow ▾
Let (X1,Y1),(X2,Y2),(X3,Y3)(X_{1}, Y_{1}), (X_{2}, Y_{2}), (X_{3}, Y_{3}) be i.i.d. bivariate normal with mean (0, 0) and correlation matrix [[1,ρ],[ρ,1]][[1, \rho], [\rho, 1]], |ρ\rho| < 1. Let Sρ=3E(S_\rho = 3E(sgn(X1−X2)(Y1−Y3))(X_{1} - X_{2})(Y_{1} - Y_{3})). Which of the following statements are true?
  1. A.If X1X_{1} and Y1Y_{1} are independent, then Sρ=0S_\rho = 0✓ Correct
  2. B.Sρ=(6/π)sin⁡S_\rho = (6/\pi) \sin⁻1(ρ/2)^{1}(\rho/2)✓ Correct
  3. C.If Sρ=0S_\rho = 0, then X1X_{1} and Y1Y_{1} are independent✓ Correct
  4. D.If X1X_{1} and Y1Y_{1} are independent, then Sρ=1/2S_\rho = 1/2

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

Part B2023 Juneshow ▾
Suppose X=(X1,X2,X3,X4)TX = (X_{1}, X_{2}, X_{3}, X_{4})^{T} ~ N4(0,I2N_{4}(0, I_{2} ⊗ ∑)\sum) with ∑=[[2,−1],[−1,2]]\sum = [[2, -1], [-1, 2]]. Define Z=[[X1,X2],[X3,X4]]Z = [[X_{1}, X_{2}], [X_{3}, X_{4}]] and Q = ZᵀZ. If Wm(n,∑)W_{m}(n, \sum) denotes a Wishart distribution of order m with n degrees of freedom, the distribution of Q11+Q12+Q21+Q22Q_{11} + Q_{12} + Q_{21} + Q_{22} is
  1. A.W1(2,2)W_{1}(2, 2)✓ Correct
  2. B.W1(1,2)W_{1}(1, 2)
  3. C.W1(2,1)W_{1}(2, 1)
  4. D.2χ422\chi^{2}_{4}

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

Part B2023 Juneshow ▾
Let X=(X1,X2)TX = (X_{1}, X_{2})^{T} be bivariate normal with mean (0, 0)ᵀ and covariance ∑=[[5,−3],[−3,10]]\sum = [[5, -3], [-3, 10]]. The mean vector and covariance matrix of Y=(X1,5−2X2)TY = (X_{1}, 5 - 2X_{2})^{T} are
  1. A.(0, 5)ᵀ, [[5, −3], [−3, 40]]
  2. B.(0, 5)ᵀ, [[5, −6], [−6, 20]]
  3. C.(0, 5)ᵀ, [[5, 3], [3, 20]]
  4. D.(0, 5)ᵀ, [[5, 6], [6, 40]]✓ Correct

Solution

E[5−2X2]=5;Var(5−2X2)=4⋅10=40;Cov(X1,5−2X2)=−2⋅(−3)=6E[5 - 2X_{2}] = 5; Var(5 - 2X_{2}) = 4\cdot10 = 40; Cov(X_{1}, 5 - 2X_{2}) = -2\cdot(-3) = 6.

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Part Cshow ▾
Let (X, Y) be bivariate normal with means 0, variances 1, correlation ρ\rho. Which of the following are true?
  1. A.X + Y and X − Y are independent.✓ Correct
  2. B.If ρ=0\rho = 0, then X and Y are independent.✓ Correct
  3. C.Given X = x, the conditional law of Y is normal with mean ρx\rho{}x and variance 1−ρ21 - \rho^{2}.✓ Correct
  4. D.For random variables in general (not jointly normal), zero correlation implies independence.

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