List of top Statistics Questions

Let $X$ be a random variable having the Poisson(4) distribution and let $E$ be an event such that $P(E|X = i) = 1 - 2^{-i}$, $i = 0, 1, 2, \ldots$. Then $P(E)$ equals ........... (round off to two decimal places).
 

Let $U \sim F_{5,8}$ and $V \sim F_{8,5}$. If $P[U > 3.69] = 0.05$, then the value of $c$ such that $P[V > c] = 0.95$ equals ................ (round off to two decimal places).
 

Let the sample mean based on a random sample from Exp($\lambda$) distribution be 3.7. Then the maximum likelihood estimate of $1 - e^{-\lambda}$ equals ........... (round off to two decimal places).
 

Let $X$ be a single observation drawn from $U(0, \theta)$ distribution, where $\theta \in \{1, 2\}$. To test $H_0: \theta = 1$ against $H_1: \theta = 2$, consider the test procedure that rejects $H_0$ if and only if $X > 0.75$. If the probabilities of Type-I and Type-II errors are $\alpha$ and $\beta$, respectively, then $\alpha + \beta$ equals ......... (round off to two decimal places).
 

Let $X$ and $Y$ be independent random variables with respective moment generating functions $M_X(t) = \dfrac{(8 + e^t)^2}{81}$ and $M_Y(t) = \dfrac{(1 + 3e^t)^3}{64}$, $-\infty < t < \infty$. Then $P(X + Y = 1)$ equals .............. (round off to two decimal places).
 

Let $x_1 = 1, x_2 = 4$ be the data on a random sample of size 2 from a Poisson($\theta$) distribution, where $\theta \in (0, \infty)$. Let $\hat{\psi}$ be the uniformly minimum variance unbiased estimate of $\psi(\theta) = \sum_{k=4}^{\infty} e^{-\theta} \dfrac{\theta^k}{k!}$ based on the given data. Then $\hat{\psi}$ equals ............ (round off to two decimal places). 
 

Let $X$ be a random variable having $N(\theta,1)$ distribution, where $\theta \in \mathbb{R}$. Consider testing $H_0: \theta = 0$ against $H_1: \theta \neq 0$ at $\alpha = 0.617$ level of significance. The power of the likelihood ratio test at $\theta = 1$ equals ............ (round off to two decimal places).
 

Let $(X, Y)$ have the joint pdf \[ f(x, y) = \begin{cases} \dfrac{3}{4}(y - x), & 0 < x < y < 2, \\ 0, & \text{otherwise.} \end{cases} \] Then the conditional expectation $E(X|Y = 1)$ equals ........... (round off to two decimal places). 
 

Consider the two probability density functions (pdfs): \[ f_0(x) = \begin{cases} 2x, & 0 \le x \le 1, \\ 0, & \text{otherwise}, \end{cases} f_1(x) = \begin{cases} 1, & 0 \le x \le 1, \\ 0, & \text{otherwise.} \end{cases} \] 

Let $X$ be a random variable with pdf $f \in \{f_0, f_1\}$. Consider testing  $H_0: f = f_0(x)$ against $H_1: f = f_1(x)$ at $\alpha = 0.05$ level of significance. For which observed value of $X$, the most powerful test would reject $H_0$? 
 

Let $X_1, X_2, \ldots, X_n$ be i.i.d. Poisson($\lambda$) random variables, where $\lambda > 0$. Define \[ \bar{X} = \frac{1}{n}\sum_{i=1}^n X_i \,\, \text{and} \,\, S^2 = \frac{1}{n - 1}\sum_{i=1}^n (X_i - \bar{X})^2. \] Then which of the following statements is/are TRUE? 
 

Let $\{X_n\}_{n \ge 1}$ be a sequence of i.i.d. random variables such that \[ P(X_1 = 0) = \frac{1}{4}, P(X_1 = 1) = \frac{3}{4}. \] Define \[ U_n = \frac{1}{n} \sum_{i=1}^n X_i \,\,\text{and} \,\, V_n = \frac{1}{n} \sum_{i=1}^n (1 - X_i)^2, n = 1, 2, \ldots \] Then which of the following statements is/are TRUE? 
 

Let $(X, Y)$ have the joint probability mass function \[ f(x, y) = \begin{cases} \binom{x + 1}{y} \binom{16}{x} \left(\frac{1}{6}\right)^y \left(\frac{5}{6}\right)^{x + 1 - y} \left(\frac{1}{2}\right)^{16}, & y = 0, 1, \ldots, x + 1; \, x = 0, 1, \ldots, 16, \\ 0, & \text{otherwise.} \end{cases} \] 

Then which of the following statements is/are TRUE? 
 

Let $X_1, X_2, \ldots, X_n$ be a random sample from $U(1,2)$ and $Y_1, Y_2, \ldots, Y_n$ be a random sample from $U(0,1)$. Suppose the two samples are independent. Define \[ Z_i = \begin{cases} 1, & \text{if } X_i Y_i < 1, \\ 0, & \text{otherwise}, \end{cases} i = 1,2, \ldots, n. \] If $\lim_{n \to \infty} P\left(|\frac{1}{n} \sum_{i=1}^n Z_i - \theta| < \epsilon\right) = 1$ for all $\epsilon > 0$, then $\theta$ equals 
 

Let $X_1, X_2, \ldots, X_n$ be a random sample from $\text{Exp}(\theta)$ distribution, where $\theta \in (0, \infty)$. If $\bar{X} = \dfrac{1}{n}\sum_{i=1}^n X_i$, then a 95% confidence interval for $\theta$ is
 

Let $X_1, X_2, \ldots, X_n$ be a random sample from $N(\mu_1, \sigma^2)$ distribution and $Y_1, Y_2, \ldots, Y_m$ be a random sample from $N(\mu_2, \sigma^2)$ distribution, where $\mu_i \in \mathbb{R}, i = 1, 2$ and $\sigma > 0$. Suppose that the two random samples are independent. Define \[ \bar{X} = \frac{1}{n} \sum_{i=1}^{n} X_i \text{and} W = \frac{\sqrt{mn} (\bar{X} - \mu_1)}{\sqrt{\sum_{i=1}^{m} (Y_i - \mu_2)^2}}. \] Then which one of the following statements is TRUE for all positive integers $m$ and $n$? 
 

Let the joint probability density function of $(X, Y)$ be \[ f(x, y) = \begin{cases} 2e^{-(x + y)}, & 0 < x < y < \infty, \\ 0, & \text{otherwise.} \end{cases} \] 

Then $P\left(X < \dfrac{Y}{2}\right)$ equals