(Gaussian Distribution) Consider a standard Gaussian random variable with probability density function given by

You may use any computer software of your choice to complete this assignment. Random variables from the four probability distributions given may be generated as follows: 1. A standard uniform random variable, U in the interval (0,1), i.e., U ~ U(0,1), may be generated using any computer software code that you have chosen. For example, in Matlab, you may use the function rand. The corresponding uniform random variable, X in the interval (-1,1) may be obtained as X = 2U 1. 2. A standard raised cosine random variable may be generated as X = Fx?(U), where Fx(x) is the CDF of the raised cosine distribution. 3. A standard Laplacian random variable may be generated as X = ln(01/02), where U7-U(0,1) and U2-0(0,1) are independent standard uniform random variables. 4. A standard Gaussian random variable Z may be generated by using your chosen software code; in Matlab, use the function 'randn'. Exercises: 1. For each of the four distributions below, compute the reliability function and the hazard function as shown (theory) in the table. 2. Generate a large number (at least 5000) random variables and estimate the reliability function and the hazard function (simulation). (a) Compute the PDF by estimating what fraction of the samples fall in the interval (x ,x + ), for a small value of . (b) Compute the CDF, P(X