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Probability and Statistics using Matlab

Task 1 Normal distribution  

Suppose that a random variable X follows a normal distribution with mean µX and standard deviation X, which you can choose. Generate N samples of X. From the samples, construct a histogram (normalized to a PDF), calculate the sample mean, standard deviation, and skewness, and calculate the empirical CDF FX(x) = P(X<x) for selected values of x. Compare these with the theoretical results. Try using different parameters, e.g. different µX, X, N, number of bins for histogram, etc (not necessarily all of these), and discuss the results (do not simply repeat the parameters presented in the slides!)

Task 2 Lognormal distribution

Next, suppose that X follows a lognormal distribution with a given mean and standard deviation. Repeat the above. In addition, you can compare the histogram with the theoretical lognormal pdf as well as the normal pdf, and discuss the difference. Try with different coefficients of variation (c.o.v.). What happens when c.o.v. is quite small (e.g. 0.05) and relatively large?

Bonus task (not compulsory)

Feel free to explore other distributions, e.g. uniform distribution, exponential distribution, Rayleigh distribution, t-distribution, chi-squared distribution. However, you will need to find out the Matlab functions yourself! (but should be quite easy. Hint: google). Generate samples from your desired distribution. From these samples, construct the normalized histogram, and compare it with the theoretical pdf similar to Tasks 1 and 2.

Task 3 Simple Reliability problem

Suppose we have a pile of capacity R and applied load S, and assume that R and S are independent. The means and standard deviations are:

pF = P(R – S < 0) ——-Eq. (1)
or equivalently
pF = P(R/S < 1) ——-Eq. (2)

Using Monte Carlo Simulation, estimate pF for 3 cases. For each case, the distribution type for R and S are specified in the following table.

How does the choice of distribution type affect pF?

Optional: For Case 1 and Case 2, it is possible to calculate pF theoretically. Hint: for Case 1, use Eq. (1). For Case 2, use Eq. (2). Compare with the simulation results. This task is optional but will be very helpful for your exam preparation!