(Solved): 4. Consider the following probability density function for a continuous random variable \( X \) : \ ...

4. Consider the following probability density function for a continuous random variable \( X \) : \[ f(x)=\left\{\begin{aligned} 0 & \text { for } x<1 \\ A \sqrt{x} & \text { for } 1 \leq x \leq 4, \\ 0 & \text { for } x>4 . \end{aligned}\right. \] (a) Find the value of the parameter \( A \) so that \( f(x) \) is a valid probability density function. (b) Calculate \( E(X) \) and \( V(X) \) (c) Find the \( 50^{\text {th }} \) percentile for this distribution (also known as the median). (d) Find the probability that the random variable \( X \) is within 2 standard deviations from the mean. In other words, find \( P(|X-\mu|<2 \sigma) \).