St303 2017 coursework the income of families in the area

Question

Q1. Use a simulation method to evaluate the function

C(a) = 0∫1 (x^-a-x^a/1+x)dx

where 0 < a < 1 for various values of a. Consider also the random variable X with density

(x^-a-x^a/C(a)(1+x))

(0 < x < 1) and evaluate E(X) for various values of a.

Q2. Generate samples X₁, X₂, . . . of independent random variables with the density in the previous question for three different values of a. Use your samples to draw histograms and compare the sample means to the values from the previous question.

Q3. Generate a sample from the distribution of the random variable

S = _i=1∑^NY_i

where Y₁, Y₂, . . . is a sequence of independent and identically distributed exponential random variables with mean 1 and N is a discrete random variable that is independent of the sequence Y₁, Y₂, . . . and such that

Pr(N = n) = (e^-4/(e-1)n!)(5ⁿ - 4ⁿ)

for n = 1, 2, . . . Use your sample to calculate (and plot!) the function

g(t) = Pr(S < tE(S))

for various values of t.

Q4. This exercise is fictitious although loosely based on an actual event. A ballet class has 21 places. A discount system is operating for the fee under which any pupil whose family income is less than 30% of the median income of the 21 families (siblings are not allowed) is entitled to a discount. The income of families in the area has a distribution with density

½e^-x + (1/4√x)e^-√x,

(x > 0) measured in appropriate units. Using Monte Carlo methods find the distribution of the number of pupils entitled to a discount.

An actuary whose child is attending the class claimed that this exercise was not entirely necessary as the distribution can be reasonably approximated by a binomial. She provided no reasoning for her claim. Is she right? Can you guess her reasoning?