Question

Test the gamma pseudo-random number generator on your computer. Simulate 10,000 gamma pseudo-random variables with parameters a and 1 for \(a = 0.5,1,1.5,2,5,\) 10. Then draw gamma quantile plots

Short answer

Generate using \(R\) with \(gamma(n,\) shape \( = 10,\) rate \(1),\) use the code for \(Q - Q\) plots.

Step-by-step solution

Pseudo-random generator

The procedures widely used to produce random values are not random. When random variables are produced by a predetermined method but pass the predefined test statistic for random chance, they are called pseudo-random numbers. The series of counts produced by a system is entirely calculated by the method's data input (or the initial pseudo-random).

To draw gamma quantile plots.

RStudio can use the built-in functions Gamma (n, Shape = 10, Rate = 1) to generate a random sample of size n = 10000 from the gamma distribution. Here, the shape is represented by. It depends on the alpha of the exercise. Then plot the Q - Q plot using the following code.

From the numbers below for the various parameter choices, we can see that the theoretical and sample quantiles generate reasonably straight lines. The sample is taken from the gamma distribution except for the points above. For, the variance of the gamma distribution is expected.