GATE Statistics Practice Quiz

Question 1

Let X_1,\dots,X_n be a random sample from N(\mu,\sigma^2), where \sigma^2 is known. Which of the following is the exact sampling distribution of the sample mean \bar X?

Accepted Answer

\bar X \sim N\left(\mu,\sigma^2/n\right)

Question 2

Suppose X_1,\dots,X_n are i.i.d. with mean \mu and variance \sigma^2<\infty. By the central limit theorem, the asymptotic distribution of \sqrt{n}(\bar X-\mu) is

Accepted Answer

N(0,\sigma^2)

Question 3

Let X_1,\dots,X_n be i.i.d. with density f(x;	heta)=	heta x^{	heta-1}, 0<x<1, 	heta>0. The MLE is \hat	heta=-\dfrac{n}{\sum_{i=1}^n \ln X_i}. Which is the exact distribution of \dfrac{n	heta}{\hat	heta}?

Accepted Answer

ext{Gamma}(n,1)

Question 4

Let X_1,\dots,X_n be i.i.d. Uniform(0,	heta), 	heta>0, and let M=X_{(n)}. Which statement about the sampling distribution of the estimator M is correct?

Accepted Answer

M/	heta \sim 	ext{Beta}(n,1)

Question 5

Let X_1,\dots,X_n be i.i.d. Uniform(0,	heta), 	heta>0. If M=X_{(n)}=\max(X_1,\dots,X_n), then the sampling density of M is

Accepted Answer

f_M(m)=\dfrac{n m^{n-1}}{	heta^n},\ 0<m<	heta

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