Random sample.

2. Suppose 42% of Ph.D.s in statistics/biostatistics are earned by woman and a random sample 36 statistics/iostatistics Ph.D.s is obtained.(You may want to use technology to compute some of the probabilities.)

a. What's the probability none of the 36 statistics/biostatistics Ph.D.s are women?

b. What's the probability at least 1 of the 36 statistics/biostatistics Lh.D.s is a woman?

c. How many woman do you expect in the sample of the 36 statistics/biostatistics Ph.D.s?

d. What's the probability fewer than half of hte 36 statistics/biostatistics Ph.D.s are women?

e. What's the probability fewer than half of the 36 statistics/biostatistics Ph.D.s are women if at least 12 are not women?

f. What's the probability the first woman is the 5th person in the sample?

g. What's the probability the first woman is the 15th person in the sample?

Generating a pair of independent Gaussian random variables: Understand the procedure in 5.10 Generating independent Gaussian random variables of the textbook. Write a Matlab code that generates a pair of independent Gaussian random variables and displays the following. You can use other programming tools available. (a) Histograms for a Gaussian random variable for 1000 and 5000 samples. (b) Scattergrams for 1000 and 5000 pairs of Gaussian random variables. What to submit: one pdf file that contains the following. Only pdf file type is allowed for upload. (25pt) Description of the method with your code (25pt) Two histogram figures for 1000 and 5000 samples. Refer to Figure 5.28 (25pt) Two scattergrams for 1000 and 5000 pairs of Gaussian random variables. Refer to Figure 5.29 (25pt) Your comments on the resultsUsing the characteristic function of the multivariate Gaussian distribution, prove that the sum of N jointly Gaussian random variables, that are not necessarily independent, is a Gaussian random variable. Obtain the mean and variance of the sum. 7. Let Y = ex, find the pdf of Y when X is a Gaussian random variable with mean m and variance o. In this case Y is said to be a lognormal random variable.'H' Hill} Exercise 6. Let X be a Gaussian random variable with mean 0 and variance oi :2 0. Let Y be a Gaussian random variable with mean 0 and variance 0% 1::- 0. Assume that X and Y are independent. Show that X + Y is also a Gaussian random variabie with mean [I and variance 0'3: + oi. (Hint: write an expression for P[X + Y E t), t E R, then take a derivative in t.)