English
Hindi

Question

Download Solution PDF

Suppose X = (X1, X2, X3, X4)T has a multivariate normal N4(0, I2 ⊗ Σ), where I2 is the 2 × 2 identity matrix, ⊗ is the Kronecker product, and Σ = . Define Z =  and Q = ((Qij)) = ZTZ. Suppose  denotes a chi-square random variate with n degrees of freedom, and Wm(n, Σ) denotes a Wishart distribution of order m with parameters n and Σ. The distribution of (Q11 + Q12 + Q21 + Q22) is

This question was previously asked in
CSIR UGC (NET) Mathematical Science: Held On (7 June 2023)
Download PDF Attempt in App
View all CSIR NET Papers >
  1. W1(2, 2)
  2. W1(1, 2)
  3. W1(2, 1)

Answer (Detailed Solution Below)

Option 1 : W1(2, 2)
Free Tests
View all Free tests >
Free
Seating Arrangement
10 Qs. 20 Marks 15 Mins
Start Now

Detailed Solution

Download Solution PDF

The Correct Answer is option 1 

We are Mainly Focused on Applied and Pure Mathematics 

We will try to Provide the solution of Statistics Part if Possible.

Download Solution PDF
Latest CSIR NET Updates

Last updated on Jul 8, 2025

-> The CSIR NET June 2025 Exam Schedule has been released on its official website.The exam will be held on 28th July 2025.

-> The CSIR UGC NET is conducted in five subjects -Chemical Sciences, Earth Sciences, Life Sciences, Mathematical Sciences, and Physical Sciences. 

-> Postgraduates in the relevant streams can apply for this exam.

-> Candidates must download and practice questions from the CSIR NET Previous year papers. Attempting the CSIR NET mock tests are also very helpful in preparation.

‹‹ Previous Ques
Next Ques ››
Exams Tests Super Practice

More Discrete & Continuous Univariate Distributions Questions

Q1.Let X = (X1, X2)T follow a bivariate normal distribution with mean vector (0, 0)T and covariance matrix Σ such that Σ = \(\left[\begin{array}{cc} 5 & -3 \\ -3 & 10 \end{array}\right]\) The mean vector and covariance matrix of Y = (X1, 5 − 2X2)T are
Q2.Suppose X = (X1, X2, X3, X4)T has a multivariate normal N4(0, I2 ⊗ Σ), where I2 is the 2 × 2 identity matrix, ⊗ is the Kronecker product, and Σ = \(\left[\begin{array}{cc}2 & -1 \\ -1 & 2\end{array}\right]\). Define Z = \(\left(\begin{array}{ll} \rm X_1 & \rm X_2 \\ \rm X_3 & \rm X_4\end{array}\right)\) and Q = ((Qij)) = ZTZ. Suppose \(\rm χ_n^2\) denotes a chi-square random variate with n degrees of freedom, and Wm(n, Σ) denotes a Wishart distribution of order m with parameters n and Σ. The distribution of (Q11 + Q12 + Q21 + Q22) is

More Statistics & Exploratory Data Analysis Questions

Q1.Let a point P be chosen at random on the line segment AB of length α. Let Z1 and Z2 denote the lengths of line segments AP and BP respectively. Then the value of E(|Z1 - Z2|) Is 
Q2.Let X1....X10 be a random sample from a distribution with the probability density function \(\rm f(x|θ)=\left\{\begin{matrix}θ x^{θ-1}, & if \:0<x<1\\\ 0, & otherwise,\end{matrix}\right.\) where θ > 0 Is an unknown parameter. The prior distribution of θ is given by   \(\rm \pi (θ)=\left\{\begin{matrix}θ e^{-θ}, & if\ θ>0\\\ 0, & otherwise,\end{matrix}\right.\) The Bayes estimator of θ under squared error loss is  
Q3.Let X1, X2 be a random sample from a population having probability density function f ∈ { f0, f1} where  \(\rm f_0(x)=\left\{\begin{matrix}\frac{1}{2}& if\ 0\le x\le 2\\\ 0&otherwise\end{matrix}\right.\ and \ \rm f_1(x)=\left\{\begin{matrix}\frac{1}{4}& if\ 0\le x\le 4\\\ 0&otherwise\end{matrix}\right.\) For testing the null hypothesis H0 : f = f0 against the alternate hypothesis H1 : f = f1, the power of a most powerful test of size α = 0.05 is equal to  
Q4.The expected number of distinct units in a simple random sample of 3 units drawn with replacement from a population of 100 units is  
Q5.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  
Q6.Let A1, A2, A3 be events satisfying 0 < P(Ai) < 1 for i = 1, 2, 3. Which of the following statements is true? 
Q7.Consider a distribution with probability mass function  \(\rm f(x| θ)=\left\{\begin{matrix}\frac{1-θ}{2}, &if\ x = 0\\\ \frac{1}{2}&if\ x=1\\\ \frac{θ}{2}&if\ x=2\\\ 0&otherwise\end{matrix}\right.\) where θ ∈ (0, 1) is an unknown parameter. In a random sample of size 100 from the above distribution, the observed counts of 0,1 and 2 are 20, 30 and 50 respectively. Then, the maximum likelihood estimate of θ based on the observed data is  
Q8.Let X0, X1 ......Xp (p ≥ 2) be independent and identically distributed random variables with mean 0 and variance 1. Suppose Yi = X0 + Xi, i = 1....p. The first principal component based on the covariance matrix of Y = (Y1...., Yp)T is
Q9.Let {Xn |n ≥ 0} be a homogeneous Markov chain with state space S = {0, 1, 2, 3, 4} and transition probability matrix  \(P=\begin{array}{r} \\ 0\\ 1\\ 2\\ 3\\ 4\end{array}\left(\begin{array}{ccccc}0&1&2&3&4\\ 1/4&0&0&3/4&0\\ 0&1&0&0&0\\ 1/3&2/3&0&0&0\\ 3/4&0&0&1/4&0\\ 1/8&1/8&1/2&1/8&1/8\end{array}\right)\) Let α denote the probability that starting with state 4 the chain will eventually get absorbed in closed class {0, 3}. Then the value of α is 

More Mathematical Science Questions

Q1.The only function among the following that satisfies Cauchy's Riemann (C-R) equations is : 
Q2.What is the cardinality of the set of real solutions of ex + x = 1?
Q3.Let A be a 10 x 10 real matrix. Assume that the rank of A is 7. Which of the following statements is necessarily true? 
Q4.Let a point P be chosen at random on the line segment AB of length α. Let Z1 and Z2 denote the lengths of line segments AP and BP respectively. Then the value of E(|Z1 - Z2|) Is 
Q5.Consider the contour γ given by \(\rm \gamma (\theta)=\left\{\begin{matrix}e^{2i \theta}&for\ \theta \in [0, \pi /2]\\\ 1+2e^{2i\theta}&for\ \theta \in [\pi/2, 3\pi/2]\\\ e^{2i\theta}&for\ \theta \in [3\pi/2, 2\pi]\end{matrix}\right.\) Then what is the value of \(\rm \int_{\gamma}\frac{dz}{z(z-2)}\)?
Q6.Let X1....X10 be a random sample from a distribution with the probability density function \(\rm f(x|θ)=\left\{\begin{matrix}θ x^{θ-1}, & if \:0<x<1\\\ 0, & otherwise,\end{matrix}\right.\) where θ > 0 Is an unknown parameter. The prior distribution of θ is given by   \(\rm \pi (θ)=\left\{\begin{matrix}θ e^{-θ}, & if\ θ>0\\\ 0, & otherwise,\end{matrix}\right.\) The Bayes estimator of θ under squared error loss is  
Q7.For a quadratic form f(x, y, z) ∈ ℝ[x, y z], we say that (a, b, c) ∈ ℝ3 is a zero of f if f(a, b, c) = 0. Which of the following quadratic forms has at least one zero different from (0, 0, 0)? 
Q8.Let A : ℝm → ℝn be a non-zero linear transformation. Which of the following statements is true? 
Q9.Let X1, X2 be a random sample from a population having probability density function f ∈ { f0, f1} where  \(\rm f_0(x)=\left\{\begin{matrix}\frac{1}{2}& if\ 0\le x\le 2\\\ 0&otherwise\end{matrix}\right.\ and \ \rm f_1(x)=\left\{\begin{matrix}\frac{1}{4}& if\ 0\le x\le 4\\\ 0&otherwise\end{matrix}\right.\) For testing the null hypothesis H0 : f = f0 against the alternate hypothesis H1 : f = f1, the power of a most powerful test of size α = 0.05 is equal to  
Q10.Let V be the real vector space of 2 x 2 matrices with entries in ℝ. Let T : V → V denote the linear transformation defined by T(B) = AB for all B ∈ V, where \(\rm A=\begin{pmatrix}2&0\\\ 0&1\end{pmatrix}\). What is the characteristic polynomial of T? 
Testbook Edu Solutions Pvt. Ltd.
1st & 2nd Floor, Zion Building,
Plot No. 273, Sector 10, Kharghar,
Navi Mumbai - 410210
support@testbook.com
Toll Free:1800 833 0800
Office Hours: 10 AM to 7 PM (all 7 days)
Company
About usCareers We are hiringTeach Online on TestbookPartnersMediaSitemap
Products
Test SeriesTestbook PassOnline CoursesOnline VideosPracticeBlogRefer & EarnBooks
Copyright © 2014-2022 Testbook Edu Solutions Pvt. Ltd.: All rights reserved
User PolicyTermsPrivacy
Hot Links： teen patti online game teen patti 500 bonus teen patti master king teen patti go