GATE Statistics Practice Quiz

Question 1

Let $X$ have probability density function $f(x)=c x^2$, $0<x<1$. What is $P(X>\frac{1}{2})$?

Accepted Answer

$\frac{7}{8}$

Question 2

Suppose $X_1,\ldots,X_n$ is a random sample from $N(\mu,\sigma^2)$, where both $\mu$ and $\sigma^2$ are unknown. Which of the following statements are correct?

Accepted Answer

$\bar X$ and $S^2=\frac{1}{n-1}\sum_{i=1}^n (X_i-\bar X)^2$ are independent.

Question 3

In a simple linear regression model $Y_i=\beta_0+\beta_1 x_i+\varepsilon_i$, $i=1,\ldots,n$, with fixed $x_i$'s and $\varepsilon_i$ independent with mean $0$ and common variance $\sigma^2$, which expression gives $\operatorname{Var}(\hat\beta_1)$?

Accepted Answer

$\frac{\sigma^2}{\sum_{i=1}^n (x_i-\bar x)^2}$

Question 4

Let $X\sim \operatorname{Poisson}(\lambda)$. Which of the following statements are correct?

Accepted Answer

The moment generating function is $M_X(t)=\exp\{\lambda(e^t-1)\}$.

Question 5

Let $X_1,\ldots,X_n$ be a random sample from the density $f(x;\theta)=\theta x^{\theta-1}$, $0<x<1$, $\theta>0$. The maximum likelihood estimator of $\theta$ is

Accepted Answer

$\hat\theta=-\frac{n}{\sum_{i=1}^n \log X_i}$

Practice Quizzes

GATE Statistics Practice Quiz