V and W are independent random variables with the following density functions:   Find the distribution function of random variable X = V + W

Let X be a random number between 0 and 1 produced by the idealized uniform random number generator. Use the density curve for X, shown below, to find the probabilities:

Automobile engines and transmissions are produced on assembly lines, and are inspected for defects after they come off their assembly lines. Those with defects are repaired. Let X represent the number of engines, and Y the number of transmissions that require repairs in a one-hour time interval. Let Z = X + Y represent the total number of repairs needed. The joint probability mass function of X and Y is as follows:   y x 0 1 2 3 0 0.13 0.10 0.07 0.03 1 0.12 0.16 0.08 0.04 2 0.02 0.06 0.08 0.04 3 0.01 0.02 0.02 0.02     Find σZ.. (Round the final answer to three decimal places.)

A standard normal random variable given by Z.

Two independent random variables are defined with the following distributions: Xp Y p 1 0.5 1 0.2 2 0.2 2 0.2 3 0.3 4 0.6 (a) Determine the PDF of the sum of X and Y using convolutions. (b) Determine the mean and variance of the sum by treating the result as the sum of two independent random variables.