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 $\sigma^2\lt \infty$. 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 $f(x;\theta)=\theta x^{\theta-1}, 0\lt x\lt 1, \theta\gt 0$. The MLE is $\hat\theta=-\dfrac{n}{\sum_{i=1}^n \ln X_i}$. Which is the exact distribution of $\dfrac{n\theta}{\hat\theta}$?

Accepted Answer

$\text{Gamma}(n,1)$

Question 4

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

Accepted Answer

$M/\theta \sim \text{Beta}(n,1)$

Question 5

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

Accepted Answer

$f_M(m)=\dfrac{n m^{n-1}}{\theta^n},\ 0\lt m\lt \theta$

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