An important distribution that arises in life is X ~ Z 2, where Z is the standard normal distribution. a. Find the support of X. b. Find E (X) algebraically. Check your answer by running a simulation involving 10000 random values of Z c. Find the median (50":l percentile) of X, and then check your answer by generating 10000 random samples of Z and seeing what percentage of their squares are less than your median. FYI: X is also said to have the chi-squared distribution with 1 degree of freedom. Fact: The 12 distribution with k degrees of freedom is the sum of the squares of k independent standard normal random variables (see HW5#2 for the case k = 1). a. Verify this in R for the case k = 5 by overlaying the histograms of 10,000 samples om each of these two descriptions. Show your code and graphs. b. If X ~ xi, nd E (X) by hand. Your answer may depend on k and you may refer to HWS