Continuous Probability Distributions - Standard Univariate Distributions - Statistics Previous Year Questions

Let \(X\) and \(Y\) be two independent random variables with \(\chi_m^2\) and \(\chi_n^2\) distributions, respectively, where \(m\) and \(n\) are positive integers. Then which of the following statements is true?

Let \(X_1, \dots, X_{20}\) be independent and identically distributed random variables with the common probability density function \(f(x) = \frac{1}{6}e^{-\frac{|x-2|}{3}}, \quad x \in (-\infty, \infty).\) Then the distribution of the random variable \(W = \frac{2}{3}\sum_{i=1}^{20} |X_i - 2|\) is

Let \( X \) be a random variable having uniform distribution on \( \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right] \). Then which one of the following statements is NOT true?

Let \(X\) and \(Y\) be two independent exponential random variables with \(E(X^2) = \frac{1}{2}\) and \(E(Y^2) = \frac{2}{9}\). Then \(P(X < 2Y)\) (rounded off to two decimal places) is equal to ______

Let \( X \) be a random variable with probability density function \[ f(x) = \begin{cases} \alpha \lambda x^{\alpha-1} e^{-\lambda x^\alpha} & \text{if } x > 0 \\ 0 & \text{otherwise}, \end{cases} \] where \( \alpha > 0 \) and \( \lambda > 0 \). If the median of \( X \) is 1 and the third quartile is 2, then \( (\alpha, \lambda) \) equals

Suppose that \( X \) has the probability density function \[ f(x) = \begin{cases} \frac{\lambda^\alpha}{\Gamma(\alpha)} x^{\alpha-1} e^{-\lambda x} & \text{if } x > 0 \\ 0 & \text{otherwise}, \end{cases} \] where \( \alpha > 0 \) and \( \lambda > 0 \). Which one of the following statements is NOT true?

Suppose that \( U \) and \( V \) are two independent and identically distributed random variables each having probability density function
\[ f(x) = \begin{cases} \lambda^2 x e^{-\lambda x} & if \ x > 0 \\ 0 & otherwise, \end{cases} \]
where \( \lambda > 0 \). Which of the following statements is/are true?

Let \( X \) be a random variable with probability density function
\[ f(x) = \begin{cases} \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)} x^{\alpha-1} (1-x)^{\beta-1} & \text{if } 0 < x < 1 \\ 0 & \text{otherwise}, \end{cases} \]
where \( \alpha > 0, \beta > 0 \). If \( E(X) = \frac{1}{3} \) and \( E(X^2) = \frac{1}{6} \), then \( \alpha + 3\beta \) equals

At a single-staff checkout counter of a supermarket store, the time taken in minutes to complete the service of a customer is a random variable having probability density function given by
\[ f(x) = \begin{cases} \frac{1}{10} e^{-\frac{x}{10}} & \text{if } x \ge 0 \\ 0 & \text{otherwise}. \end{cases} \]
When you arrive at the counter, you observe that there is already one person in service. You are also informed that the person has been in service for 5 minutes. Assuming that the service times of different customers are independent of each other, the probability that your total waiting time (which is the sum of your waiting time in queue and service time) is more than 15 minutes equals

Let \(X\) be a continuous random variable with probability density function
\(f(x) = \frac{1}{\sigma x \sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{\log_e x - \mu}{\sigma}\right)^2},\quad x > 0,\)
where \(\mu \in \mathbb{R}, \sigma > 0\). If \(\log_e\left(\frac{E(X^2)}{(E(X))^2}\right) = 4\), then Var(\(\log_e X\)) equals