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Discrete Stochastic Processes and Optimal Filtering by Roger Ceschi, Jean-Claude Bertein

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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 ...

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