Final - Introductory Statistics - F24.docx

Introductory Statistics

1. STAT2910 – Probability and Mathematical Statistics I
Fall 2024
Final Exam
Date: Thursday 2/1/2025 Instructor Name: Dr. Amal Mohamed
Name: _________________________ ID: ______________________
Instructions:
1-The Academic Integrity code applies to this test.
2-You have 180 minutes to finish the exam!
Question Possible points Score
Question 1 6
Question 2 9
Question 3 10
Question 4 16
Question 5 10
Question 6 6
Question 7 10
Question 8 6
Question 9 6
Question 10 16
Total
Worth 35%
90
BONUS
1%
10

2. Q1. [6 pts.] An environmental engineer measures the amount (by weight) of particulate
pollution in air samples of a given volume collected over the smokestack of a coal
operated power plant. X denotes the amount of pollutant per sample collected when a
cleaning device on the stack is not in operation, and Y denotes the same amount when the
cleaning device is operating. It is known that the joint probability density function for X
and Y is
a)Find k,
b)Find the marginal densities for X and Y.
c)Find the probability that the amount of pollutant with the cleaning device in
operation is at most of the amount without the cleaning device in operation.

3. Q2. [4 pts.] A. According to recent census figures, the proportions of adults (persons over
18 years of age) in the United States associated with five age categories are as given in
the following table.
Age Proportion
18−24.18
25−34.23
35−44.16
45−64.27
65↑ .16
If these figures are accurate and five adults are randomly sampled, find the probability
that the sample contains one person between the ages of 18 and 24, two between the ages
of 25 and 34, and two between the ages of 45 and 64.
B. [5 pts.] Assume that Y
1
,Y
2, and Y
3 are random variables, with
E(Y
1) ¿2, E(Y
2) ¿−1, E(Y
3) ¿4,
V(Y
1) ¿4,V(Y
2) ¿6, V(Y
3) ¿8,
Cov⁡(Y
1,Y
2)¿1,Cov⁡(Y
1,Y
3)¿−1,Cov⁡(Y
2,Y
3)¿0.
Find E(3Y
1
+4Y
2
−6Y
3) and V(3Y
1
+4Y
2
−6Y
3).

5. Q3. [10 pts.] An engineering college has made a study of the grade-point averages of
graduating engineers, denoted by the random variable Y. It is desired to study these as
a function of high school grade-point averages, denoted by the random variable X.
The joint probability distribution is shown, where the grade point averages have been
combined into live categories for each variable.
X
2.02.53.03.54.0
2.00.050 0 0 0
2.50. 100.0400.010
Y3.00.020.100.050.100.01
3.50 00.100.200.10
4.00 00.050.020.05
a)Find the marginal distributions for X and Y.
b)Find E(X) and E(Y).
c)Find P(X  3, Y  3).
d)Find the covariance of X, and Y and comment on the direction of the relationship.

6. Q4. [16 pts.] Use any method you prefer for the problems below:
(a) Suppose that Z has a standard normal distribution. Find the density function of the random variable
U=Z
2
.
(b) Suppose that
Z
1 and
Z
2 are independent, standard normal random variables. Find the density function
of U=Z
1
2
+Z
2
2
.
(c) Let Z have a uniform distribution on (0,1). Show that W=−2ln⁡(Z) has an exponential distribution.
(d) Suppose Y has a Gamma distribution with α and β. Express the distribution of
U=
2
β
Y
.

7. Q5. (10 pts) A. 4pts. A geologist collected 10 specimens of basaltic rock and 10 specimens of granite,
and then she randomly selected 15 of the specimens for analysis.
a. Write the probability function p(x), of the number of granite specimens selected?
b. What is the probability that all specimens of one of the two types of rock are selected?
B. 3pts. If the number of calls to a fire department, Y, in a day has a Poisson distribution with mean
λ=5.3, what is the most likely number of phone calls to the fire department on any day, i.e., what is the
mode of Y?
C. 3 pts. The probability for a child exposed to a certain disease to catch it is 0.2.
a. What is the probability that among 12 children exposed to the disease 3 will catch it?
b. What is the probability that the 12
th 
child exposed to the disease will be the 1
st 
to catch it?
c. What is the probability that the 12
th 
child exposed to the disease will be the 3
rd 
to catch it?

8. Q6. (6 pts.) Suppose that X and Y are continuous random variables. Suppose that X has
an exponential distribution with parameter β = 1. That is the marginal probability
density function of X is given by:
fX(x) = {
e
−x
0≤x<¥
0 otherwise
Suppose the conditional distribution of Y given X = x is uniform from 0 to x. That is,
fY|X(y|x) =
(a)Find the joint probability distribution of X and Y.
(b)Find the conditional distribution of X given Y = y.

9. Q7. (10 pts.) Suppose that Y
1 and Y
2 are independent exponentially distributed random
variables, both with mean β, and define U
1
=Y
1
+Y
2 and U
2
=Y
1
/Y
2.
a. Show that the joint density of (U
1,U
2) is
f
U
1
,U
2
(u
1,u
2)={
1
β
2
u
1e
−u
1
/β1
(1+u
2)
2
,0<u
1,0<u
2,
0,  otherwise. 
b. Are U
1 and U
2 are independent? Why?

10. Q8. (6 pts.) Let
Y
1 and
Y
2 be independent normally distributed random variables with means
μ
1 and
μ
2, respectively, and variances σ
1
2
=σ
2
2
=σ
2
.
1. Show that
Y
1 and
Y
2 have a bivariate normal distribution with ρ=0.
2. Consider
U
1=Y
1+Y
2 and
U
2=Y
1−Y
2. Show that
U
1 and
U
2 have a bivariate normal
distribution and that
U
1 and
U
2 are independent.
3. What are the marginal distributions of
U
1 and
U
2 ?

11. Q9. (6 pts.) Suppose X has the distribution function
d)Find a.
e)Find P(X  1)
f)Find f(x) the density function of X.

12. Q10. (16 pts.) A very erratic golfer has a probability of .70 that his tee shot will go at
least 100 yards.
a)In the next three holes what is the probability that all of his tee shots will go at
least 100 yards.
b)In the next four holes what is the probability that at least three of his tee shots will
go at least 100 yards.
c)In the next four holes what is the probability that at most two of his tee shots will
go at least 100 yards.
d)In the next five holes what is the probability that more of his tee shots will go at
least 100 yards than the number of tee shots that will not go at least 100 yards.
e)What is the probability that his first tee shot that goes at least 100 yards occurs on
the fifth hole.
f)What is the probability that his first tee shot that goes at least 100 yards does not
occur before the 3rd hole.
g)What is the probability that his third tee shot that goes at least 100 yards occurs on
the fifth hole.
h)What is the probability that he does not have three tee shots that go at least 100
yards by the 5th hole.

13. BONUS 1%
A.The moment generating function for a random variable
Y
is
M
y(t)=¿
.
Expand M
y(t) in a power series in t and find μand σ
2
.
B.Suppose that X is uniform on [-1, 2]. Find the probability density function for Y=|
X|·

14. DRAFT