Let X be an r.v. of distribution function

Calculate the probabilities:

Given the random vector Z = ( X, Y) of probability density where K is a real constant and where , determine the constant K and the densities f _X and f _Y of the r.v. X and Y.

Let X and Y be two independent random variables of uniform density on the interval [0,1]:

1. Determine the probability density f _Z of the r.v. Z = X + Y;

2. Determine the probability density f _U of the r.v. U = X Y.

Let X and Y be two independent r.v. of uniform density on the interval [0,1].

Determine the probability density f _U of the r.v. U = X Y.

Solution 1.4.

U takes its values in [0,1]

Let F _U be the distribution function of U:

• if u ? 0 F _U ( u) = 0; if u ? 1 F _U( u) = 1;

• if u ?]0,1[: F _U ( u) = P ( U ? u) = P( X Y ? u) = P(( X, Y) ? B _u);

where B _u = A ? B