**1.7. Exercises for Chapter 1**

**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

*f*of the r.v.

_{Y}*X*and

*Y*.

**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.

*Z*=

*X*+

*Y*;

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

*U*=

*X Y*.

**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.

*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

where is the cross-hatched area of the figure.

Thus

Finally

**Exercise 1.5.**

Under consideration ...

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