List of top Statistics Questions on Cumulative Distribution asked in GATE ST

Let \( X \) be a random variable with distribution function \( F \), such that \[ \lim_{h \to 0^-} F(3 + h) = \frac{1}{4} \quad {and} \quad F(3) = \frac{3}{4}. \] {Then } \( 16 \, {Pr}(X = 3) \) equals ___________ (answer in integer).

Let \( X \) be a random variable with the distribution function \[ F(x) = \begin{cases} 0 & \text{if } x < 0, \\ \alpha(1 + 2x^2) & \text{if } 0 \leq x < 1, \\ 1 & \text{if } x \geq 1, \end{cases} \] where \( \alpha \) is a real constant. If the median of \( X \) is \( \frac{1}{\sqrt{2}} \), then the value of \( \alpha \) equals:

Let \( \Phi(.) \) denote the cumulative distribution function of a standard normal random variable. If the random variable \( X \) has the cumulative distribution function \[ F(x) = \begin{cases} \Phi(x) & \text{if } x < -1, \\ \Phi(x + 1) & \text{if } x \geq -1, \end{cases} \] then which one of the following statements is true?