4.3 Consider two uniform independent random variables X, Y in the range 1 to 1.

(a) Determine the distribution function, mean and variance, and the moment generating function of the variables.

(b) We speculate that the sum of the two random variables is distributed like a

“triangular” distribution between the range 2 to 2, with distribution function f .x/ D 8ˆ<

ˆ:

1 2 C x

4 if 2 x

0 1

2 

x 4

if 0 x

2 Using the moment generation function, prove that the variable Z D X C Y is distributed like the triangular distribution above.