## 4.3. Exercises

EXERCISE 4.4.

Suppose that the probability of measuring *k* points of a random process *X* within an interval *T*, is *P _{k}* =

*c*

*a*, for

^{k}*k*≥ 0.

1. Determine the value of the constant *c* as a function of parameter *a*, which belongs to an interval to be specified.

2. Demonstrate that *E[K]* = a/(1-a).

3. Calculate the probability of having at least *k* points within the interval *T*.

4. The following results are issued from process observations:

- Give an estimation of
*a*. - Verify if the measured values obey the distribution defined above.
- Explain why these values are trustworthy or not.

EXERCISE 4.5.

In order to control the homogenity of a factory production, 1,000 samples of a manufactured mechanical part are randomly selected. The size *X* measured for each of them is recorded in the table below:

The problem is to know if the distribution of *X* can be considered as Gaussian.

1. Calculate the empirical mean value, variance and standard deviation.

2. Plot the histogram and conclude.

3. For a Gaussian variable, what is the probability of having a value up to 2σ from its mean value?

4. Give a probability for 4 ≤ X ≤ 4.2 and conclude.

5. Validate your conclusion using one or two statistical tests.

EXERCISE 4.6.

The following values are recorded with a receiver:

1. Sort these values in increasing ...

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