Mathematical statistics question

MATH Introduction to Mathematical Statistics

Instructions: Complete the following. Show all work. You can submit your answers on separate paper so that you do not need to download and print this exam.

(1) (10 points) Suppose the following is the PDF of a random variable X:

f (x)  cex / 2,

where c is an unknown constant.

x  0

a. (5 points) What is the value of c that makes this a valid PDF?

b. (5 points) What is the CDF for this value of c?

(2) (10 points) Suppose the following is the PDF of a random variable X:

c. (5 points) Find E(X).

d. (5 points) Find Var(X).

f (x)  2x,

0  x  1

(3) (10 points) Consider a class of three students who scores on a statistics exam are 75, 80, 85. Let 𝑋̅ 2 denote the sample mean of scores on the exam for samples of size

𝑛 = 2. What is the sampling distribution of 𝑋̅ 2

(4) (15 points) Consider the function (k is a constant)

P ( X  x )   x k ,



0,

x  1,2

else

a. (5 points) What value of k would make this a valid joint probability mass function (PMF)?

b. (5 points) Find E[X] using this value of k.

c. (5 points) Find Var(X) using this value of k.

(5) (10 points) The following is a histogram of the grades of thirty students on a statistics exam.

a. (5 points) What is the sample proportion of students who scored between 75 and 85?

b. (5 points) Which would you use as a measure of center for this dat, the sample mean or the sample median, and why?

(6) (15 points) A chromosome mutation linked to colorblindness is known to occur once in every 10,000 births. Suppose 20,000 babies are born this year in a particular city. The occurrence of the mutation in one baby is independent of the occurrence in another baby.

a. (5 points) How many of the babies are expected to have the mutation?

b. (5 points) What is the probability exactly 1 of the 20,000 babies has the mutation?

c. (5 points) What is the probability the 5th baby surveyed is the first to have the mutation?

(7) (20 points) The Stanford-Binet IQ test is scaled to follow a Normal distribution with a mean of 100 and a standard deviation of 15.

a. (5 points) What is the probability a student scores between 85 and 115?

b. (5 points) Students who score between 80 and 120 are considered “average”. What % of students are “average”?

c. (5 points) Students who score above 130 are considered “gifted”. What % of students are “gifted”?

d. (5 points) A score of 160 or above is considered a “genius” level IQ. What

% of students score at 160 or above?