1.7. Exercises for Chapter 1

Exercise 1.1.

Let X be an r.v. of distribution function

images

Calculate the probabilities:

images

Exercise 1.2.

Given the random vector Z = (X,Y) of probability density image where K is a real constant and where image, determine the constant K and the densities fX and fY of the r.v. 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 fZ of the r.v. Z = X + Y;

2) Determine the probability density fU 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 fU of the r.v. U = X Y.

Solution 1.4.

image

U takes its values in [0,1]

Let FU be the distribution function of U:

– if u ≤ 0 FU (u) = 0; if u ≥ 1 FU (u) = 1;

– if image

where is the cross-hatched area of the figure.

Thus

Finally

Exercise 1.5.

Under consideration ...

Get Discrete Stochastic Processes and Optimal Filtering now with O’Reilly online learning.

O’Reilly members experience live online training, plus books, videos, and digital content from 200+ publishers.