List of top Statistics Questions

Let \( X \) be a continuous random variable with the probability density function
\[ f(x) = \begin{cases} \frac{x}{8}, & \text{if} \, 0 < x < 2, \\ k, & \text{if} \, 2 \leq x \leq 4, \\ \frac{6 - x}{8}, & \text{if} \, 4 < x < 6, \\ 0, & \text{otherwise}. \end{cases} \] Then \( P(1 < X < 5) \) equals

Let \( X \) and \( Y \) be discrete random variables with \( P(Y \in \{0, 1\}) = 1 \), \[ P(X = 0) = \frac{3}{4}, \, P(X = 1) = \frac{1}{4}, \, P(Y = 1 | X = 1) = \frac{3}{4}, \, P(Y = 0 | X = 0) = \frac{7}{8}. \] Then \( 3P(Y = 1) - P(Y = 0) \) equals

Let \( X \) be a random variable with the probability density function \[ f(x | r, \lambda) = \frac{x^{r-1} e^{-x/\lambda}}{\lambda^r (r-1)!}, \quad x>0, \, \lambda>0, \, r>0. \] If \( E(X) = 2 \) and \( \text{Var}(X) = 2 \), then \( P(X<1) \) equals

Let a unit vector \( \mathbf{v} = (v_1, v_2, v_3)^T \) be such that \( A\mathbf{v} = 0 \), where
\[ A = \begin{pmatrix} \frac{5}{6} & \frac{1}{3} & -\frac{1}{6} \\ \frac{1}{3} & \frac{1}{3} & -\frac{1}{3} \\ -\frac{1}{6} & \frac{1}{3} & \frac{5}{6} \end{pmatrix}. \] Then the value of \( \sqrt{6} \left( |v_1| + |v_2| + |v_3| \right) \) equals

Let \( u_n = \frac{18n + 3}{(3n - 1)^2 (3n + 2)^2} \), \( n \in \mathbb{N} \). Then \[ \sum_{n=1}^{\infty} u_n \quad \text{equals} \]

Let \( X \) and \( Y \) be continuous random variables with the joint probability density function
\[ f(x,y) = \begin{cases} \frac{1}{2} e^{-x}, & \text{if} \, |y| \leq x, \, x > 0, \\ 0, & \text{otherwise}. \end{cases} \] Then \( E(X | Y = -1) \) equals

Let \( X_1, X_2, \dots, X_{100} \) be i.i.d. random variables with \( E(X_1) = 0, E(X_1^2) = \sigma^2 \), where \( \sigma>0 \). Let \[ S = \sum_{i=1}^{100} X_i. \, \text{If an approximate value of} \, P(S \leq 30) \, \text{is} \, 0.9332, \, \text{then} \, \sigma^2 \, \text{equals}. \]

For \( j = 1, 2, \dots, 5 \), let \( P_j \) be the matrix of order \( 5 \times 5 \) obtained by replacing the \( j^{th} \) column of the identity matrix of order \( 5 \times 5 \) with the column vector \( v = (5, 4, 3, 2, 1)^T \). Then the determinant of the matrix product \( P_1 P_2 P_3 P_4 P_5 \) is

Let \[ F(x) = \int_0^x e^{t^2 - 3t - 5} \, dt, \quad x>0. \] Then the number of roots of \( F(x) = 0 \) in the interval \( (0, 4) \) is

Let \( X \) and \( Y \) be discrete random variables with the joint probability mass function \[ p(x, y) = \frac{1}{25} (x^2 + y^2), \quad \text{if} \, x = 1, 2; \, y = 0, 1, 2. \] Then \( P(Y = 1 | X = 1) \) equals

Let \( f : [0, \frac{\pi}{2}] \to \mathbb{R} \) be defined as \[ f(x) = ax + \beta \sin x, \] where \( a, \beta \in \mathbb{R} \). Let \( f \) have a local minimum at \( x = \frac{\pi}{4} \) with \[ f' \left( \frac{\pi}{4} \right) = -\frac{4}{\sqrt{2}}. \] Then \( 8\sqrt{2a} + 4 \beta \) equals

Let the random variable \( X \) have uniform distribution on the interval \( (0, 1) \) and \( Y = -2 \log X \). Then \( E(Y) \) equals

Let \( X_1, X_2, X_3, X_4 \) be a random sample from an \( N(\mu, \sigma^2) \) distribution. Let \( \bar{X} = \frac{1}{4} \sum_{i=1}^4 X_i \) and \[ \tilde{Y} = \frac{15}{7} \sum_{i=1}^4 X_i. \] If \( \bar{X} \) has a t-distribution, then \( (\nu - k) \) equals

Let \( X_1, X_2, X_3, X_4 \) be a random sample from an \( N(\theta, 1) \) distribution, where \( \theta \in (-\infty, \infty) \). Suppose the null hypothesis \( H_0: \theta = 1 \) is to be tested against the hypothesis \( H_1: \theta<1 \) at \( \alpha = 0.05 \) level of significance. For what observed values of \( \sum_{i=1}^4 X_i \), the uniformly most powerful test would reject \( H_0 \)?

Let \( E \) and \( F \) be two events with \( 0<P(E)<1 \), \( 0<P(F)<1 \) and \( P(E) + P(F) \geq 1 \). Which of the following statements is (are) TRUE?

The cumulative distribution function of a random variable \( X \) is given by
\[ F(x) = \begin{cases} 0, & \text{if } x < 0 \\ \frac{4}{9}, & \text{if } 0 \leq x < 1 \\ \frac{8}{9}, & \text{if } 1 \leq x < 2 \\ 1, & \text{if } x \geq 2 \end{cases} \] Which of the following statements is (are) TRUE?

Let \( X_1, X_2 \) be a random sample from a distribution with the probability mass function
\[ f(x|\theta) = \begin{cases} 1 - \theta, & \text{if } x = 0 \\ \theta, & \text{if } x = 1 \\ 0, & \text{otherwise}, \quad 0 < \theta < 1. \end{cases} \] Which of the following is (are) unbiased estimator(s) of \( \theta \)?

Let \( X_1, X_2, X_3 \) be a random sample from a distribution with the probability density function
\[ f(x|\theta) = \begin{cases} \frac{1}{\theta} e^{-x/\theta}, & \text{if } x > 0, \, \theta > 0 \\ 0, & \text{otherwise} \end{cases} \] If \( \hat{X}(X_1, X_2, X_3) \) is an unbiased estimator of \( \theta \), which of the following CANNOT be attained as a value of the variance of \( \hat{X} \) at \( \theta = 1 \)?

Let \( X_1, X_2, \dots, X_n \) (where \( n \geq 2 \)) be a random sample from a distribution with the probability density function
\[ f(x|\theta) = \begin{cases} \frac{x}{\theta^2} e^{-x/\theta}, & x > 0, \, \theta > 0 \\ 0, & \text{otherwise} \end{cases} \] Let \( \bar{X} = \frac{1}{n} \sum_{i=1}^n X_i \). Which of the following statistics is (are) sufficient but NOT complete?

Let \( f(x) \) be a nonnegative differentiable function on \( [a, b] \subset \mathbb{R} \) such that \( f(a) = 0 = f(b) \) and \( |f'(x)| \leq 4 \). Let \( L_1 \) and \( L_2 \) be the straight lines given by the equations \[ y = 4(x - a) \quad \text{and} \quad y = -4(x - b), \text{ respectively. Then which of the following statements is (are) TRUE?} \]

Let \( \mathbf{v} \in \mathbb{R}^k \) with \( \mathbf{v}^T \mathbf{v} \neq 0 \). Let \[ P = I - 2 \frac{ \mathbf{v} \mathbf{v}^T }{ \mathbf{v}^T \mathbf{v} }, \] where \( I \) is the \( k \times k \) identity matrix. Then which of the following statements is (are) TRUE?

Let \( \{a_n\} \) and \( \{b_n\} \) be sequences of real numbers such that \( \{a_n\} \) is increasing and \( \{b_n\} \) is decreasing. Under which of the following conditions, the sequence \( \{a_n + b_n\} \) is always convergent?

Let \( f: [0,1] \to [0,1] \) be defined as follows:
\[ f(x) = \begin{cases} x, & \text{if } x \in \mathbb{Q} \cap [0,1] \\ x + \frac{2}{3}, & \text{if } x \in \mathbb{Q}^c \cap (0, \frac{1}{3}) \\ x - \frac{1}{3}, & \text{if } x \in \mathbb{Q}^c \cap (\frac{1}{3}, 1] \end{cases} \] Which of the following statements is (are) TRUE?

Let \( x(t), y(t), 1 \leq t \leq \pi \), be the curve defined by \[ x(t) = \int_1^t \frac{\cos z}{z^2} \, dz \quad \text{and} \quad y(t) = \int_1^t \frac{\sin z}{z^2} \, dz. \] Let \( L \) be the length of the arc of this curve from the origin to the point \( P \) on the curve at which the tangent is perpendicular to the x-axis. Then \( L \) equals

Let \( Q, A, B \) be matrices of order \( n \times n \) with real entries such that \( Q \) is orthogonal and \( A \) is invertible. Then the eigenvalues of \( Q^T A^{-1} B Q \) are always the same as those of