Suppose that the probability of measuring k points of a random process X within an interval T, is Pk = c ak, for 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.
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.
The following values are recorded with a receiver:
1. Sort these values in increasing ...