Question

A fair coin is flipped independently until the first Heads is observed. Let the random variable K be the number of tosses until the first Heads is observed plus 1. For example, if we see TTTHTH, then K=5 . For k=1,2,…,K , let Xk be a continuous random variable that is uniform over the interval [0,5] . The Xk are independent of one another and of the coin flips. Let X=∑Kk=1Xk . Find the mean and variance of X . You may use the fact that the mean and variance of a geometric random variable with parameter p are 1/p and (1−p)/p2 , respectively.

What is :

E[X] = ?

Var [X] = ?