Suppose \( X \) and \( Y \) are random variables. The conditional expectation of \( X \) given \( Y \) is denoted by \( E[X | Y] \…

Let \( X = aZ + b \), where \( Z \) is a standard normal random variable, and \( a, b \) are two unknown constants. It is given th…

It is given that \( P(X \geq 2) = 0.25 \) for an exponentially distributed random variable \( X \) with \( E[X] = \frac{1}{\lambda…

There are three boxes containing white balls and black balls. Box-1 contains 2 black and 1 white ball. Box-2 contains 1 black and …

Let \( Y = Z^2 \), \( Z = \frac{X - \mu}{\sigma} \), where \( X \) is a normal random variable with mean \( \mu \) and variance \(…

Consider the cumulative distribution function (CDF) of a random variable \( X \): \[ F_X(x) = \begin{cases} 0 & if x \leq -1 \frac…