Setup for questions:

A random variableX is generated as follows. We flip a coin. With probabilityp, the result is Heads, and thenXis generated according to a PDF?_X|H which is uniform on[0, 1].

With probability1?p the result is Tails, and thenXis generated according to a PDF?_X|T

of the form:

?_X|T(x)=2x, ifx?[0,1]

(The PDF is zero everywhere else.)

Question 1: What is the (unconditional) PDF?_X(x) ofX?

For0 ? x ? 1:

?_X(x) = ?????

Question 2:

Calculate?[X]

.

?[X]= ???????

Screenshot for clarity:

A random variable X is generated as follows. We flip a coin. With probability p, the result is Heads, and then X is generated according to a PDF fx|H which is uniform on [0, 1]. With probability 1 - p the result is Tails, and then X is generated according to a PDF fxIT of the form fxIT (x) = 2x, if x E [0, 1 ] . (The PDF is zero everywhere else.) 1. What is the (unconditional) PDF fx (x) of X? For 0