The Rayleigh distribution is a continuous probability distribution for nonnegative random vari- ables. A random variable (RV) X has a Rayleigh distribution with scale parameter > 0 if it has probability density function (pdf) x fx(x) = 22 e 202 02 " x 0. We say that X ~ Rayleigh(). The cumulative distribution function for Rayleigh distribution is 2 = Fx(x) 1 e 202 NOTE: You will not receive credit if you use the drayleigh, prayleigh, qrayleigh, or rrayleigh functions. Please write your own functions. (a) (2 points) Write an R function that calculates the pdf for a Rayleigh distribution. Make sure that you add comments to the R code to describe the arguments to the function, and explain how your code works. (b) (3 points) Produce a single plot comparing the probability density function (pdf) when = = 0.5, = 1 and = 2. Label both x-axis and y-axis properly. Make sure that your figure has a legend to differentiate the values of . Interpreting this figure, what features of the probability density does the scale parameter control? (c) (4 points) Explain using mathematics and words how to generate a Rayleigh distributed random variables using the inverse transformation method. Next write an R function that generates n Rayleigh distributed random variates using this method. Make sure that you add comments to the R code in your function to describe the arguments to the function, and explain how your code works. (d) (3 points) Note that if X and Y are independent N(0,02) RVs, then ~ Rayleigh() Use this fact to write another R function that generates n Rayleigh distributed random vari- ates. Make sure that you add comments to the R code to describe the arguments to the function, and explain how your code works.