**Exercise 1.1.**

Let *X* be an r.v. of distribution function

Calculate the probabilities:

**Exercise 1.2.**

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

**Exercise 1.3.**

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.

2) Determine the probability density *f _{U}* of the r.v.

**Exercise 1.4.**

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.

*Solution 1.4.*

*U* takes its values in [0,1]

Let *F _{U}* be the distribution function of

– if *u* ≤ 0 *F _{U}* (

– if

where is the cross-hatched area of the figure.

Thus

Finally

**Exercise 1.5.**

Under consideration ...

Start Free Trial

No credit card required