Charles Xavier seeks to study the prevalence of the mutant gene among the human population. To do so, he enlists Wolverine2 to help him in collecting a sample of participants that would be tested to see if they carry the gene (and hence have superpowers). Wolverine decides to do this by picking a random sample of size 75 without replacement from the population of the 2000 people living in his hometown of Cold Lake, Alberta. Assume that exactly 300 of the citizens of Cold Lake carry the gene.

A) Let X denote the number of participants that carry the gene in Wolverine's sample.

(i) What is the distribution of X? Be sure to specify all of the parameters of the distribution.

(ii) Using R, compute P (15 X 40).

(iii) Using R, plot the probability function of X.

(iv) Using R, plot the cumulative distribution function of X. (v) [3 marks] Using R, compute the 25th, 50th, and 75th percentiles of X.

Charles also enlisted the help of Deadpool. Deadpool collected his own sample of participants for this study. However, Deadpool is a bit more careless and instead of sampling without replacement, he sampled 75 people with replacement from the town of Cold Lake and merely recorded the total number of positive test results. Recall that there are 2000 people living in Cold Lake and that exactly 300 of them carry the gene.

B) Let Y denote the number of positive test results from Deadpool's sample.

(i) What is the distribution of Y? Be sure to specify all of the parameters of the distribution.

(ii) Using R, compute P (15 Y 40).

(iii) Using R, plot the probability function of Y.

(iv) Using R, plot the cumulative distribution function of Y.

(v) Using R, compute the 25th, 50th, and 75th percentiles of Y.

c) Based on your findings in parts a) and b), what can you say comparatively about the probability distributions of X and Y? Do they appear to be similar? Should we expect them to be similar? Why or why not? Explain.