List of top Statistics Questions

Let \( X \) be a random variable with the probability density function \[ f(x) = \begin{cases} \dfrac{\alpha^{p}}{\Gamma(p)} e^{-\alpha x} x^{p-1}, & x \geq 0, \, \alpha > 0, \, p > 0, \\ 0, & \text{otherwise}. \end{cases} \] If \( E(X) = 20 \) and \( \text{Var}(X) = 10 \), then \( (\alpha, p) \) is 
 

Let \( X \) be a random variable with the distribution function \[ F(x) = \begin{cases} 0, & x < 0, \\ \frac{1}{4} + \frac{4x - x^2}{8}, & 0 \leq x < 2, \\ 1, & x \geq 2. \end{cases} \] Then \[ P(X = 0) + P(X = 1.5) + P(X = 2) + P(X \geq 1) \] equals 
 

Let \( X_1, X_2 \) and \( X_3 \) be i.i.d. \( U(0,1) \) random variables. Then \( E\left( \dfrac{X_1 + X_2}{X_1 + X_2 + X_3} \right) \) equals 
 

Let \( X \) be a random variable with the moment generating function \[ M_X(t) = \frac{6}{\pi^2} \sum_{n \geq 1} \frac{e^{t^2/2n}}{n^2}, t \in \mathbb{R}. \] Then \( P(X \in \mathbb{Q}) \), where \( \mathbb{Q} \) is the set of rational numbers, equals 
 

Let \( \{X_n\}_{n \geq 1} \) be a sequence of independent random variables with \( X_n \) having the probability density function as \[ f_n(x) = \begin{cases} \dfrac{1}{2^{n/2} \Gamma\left( \frac{5}{2} \right)} e^{\tfrac{-x^2}{2}} x^{(\tfrac{n}{2}-1)} , & x > 0, \\ 0, & \text{otherwise}. \end{cases} \] Then \[ \lim_{n \to \infty} \left[ P(X_n > \frac{3n}{4}) + P(X_n > n + 2 \sqrt{2n}) \right] \] equals 
 

Let \( X \) and \( Y \) have the joint probability density function \[ f(x, y) = \begin{cases} 2, & 0 \leq x \leq y \leq 1, \\ 0, & \text{otherwise}. \end{cases} \] Let \( a = E(Y|X = \frac{1}{2}) \) and \( b = \text{Var}(Y|X = \frac{1}{2}) \). Then \( (a, b) \) is 
 

Let \( X \) and \( Y \) have the joint probability mass function \[ P(X = m, Y = n) = \begin{cases} \frac{m + n}{21}, & m = 1, 2, 3; n = 1, 2, \\ 0, & \text{otherwise}. \end{cases}\] Then \( P(X = 2 | Y = 2) \) equals 
 

Let \( X \) and \( Y \) have the joint probability density function \[ f(x, y) = \begin{cases} e^{-y}, & 0 < x < y < \infty, \\ 0, & \text{otherwise}. \end{cases} \] Then the correlation coefficient between \( X \) and \( Y \) equals 
 

Let \( X \) be a random sample from a discrete distribution with the probability mass function \[ f(x; \theta) = P(X = x) = \begin{cases} \frac{1}{\theta}, & x = 1, 2, \dots, \theta, \\ 0, & \text{otherwise}, \end{cases} \] where \( \theta \in \{20, 40\} \) is the unknown parameter. Consider testing \[ H_0: \theta = 40 \,\, \text{against} \,\, H_1: \theta = 20 \] at a level of significance \( \alpha = 0.1 \). Then the uniformly most powerful test rejects \( H_0 \) if and only if