Question

Question 11-In a large statistics class. PM 100% of all students are male and the rest are female. The heights of all male students follow a normal distribution with mean Ax and standard deviation σΜ. The heights of all female students follow a normal distribution with mean Mr and standard deviation σ Suppose a random sample of n students is selected independently. Let H be the mean height of these n randomly selected students. a. Determine E[H) and Va[FI / NM NF Hin H-- i-1 j-1 X, is the height of the ith male student in the sample Y is the height of the jth female student in the sample where NM is the number of male students in the sample N, (-n-NM) is the number of female students in the sample · Determine the probability density function of H, fa(h) Note: Express the probability density function as an appropriate integral or series. b. c. Determine the moment-generating function of H

Answer 1

a) Given the random variable $\overline{H}=\frac{1}{n}\left ( \sum_{i=1}^{N_M}X_i+ \sum_{j=1}^{N_F}Y_i\right )$

The expectation is

NM NF E (Xi) +> E(Yi) i-1 j-1 NM NF i-1

The variance is

NM Var (H)- 1 Σ Var (X) + Σ Var(W) MOM Var (H - PM 2 1 - pM 2

b) The distribution of $\overline{H}=\frac{1}{n}\left ( \sum_{i=1}^{N_M}X_i+ \sum_{j=1}^{N_F}Y_i\right )$ is normal with,
PM2 1 PM 2
since
Xi, Yi
are normally distributed.

The PDF is

(h 2σ

c) The moment generating function of normal distribution is $e^{\mu t+\sigma ^2t^2/2}$ . Thus,

${\color{Blue} M_{\overline{H}}\left ( t \right )=e^{\mu_{\overline{H}} t+\sigma ^2_{\overline{H}}t^2/2}}$