1. Introduction to Probability: Solutions for Quizzes 4 and 5
Suhan Yu
Department of Computer Science & Information Engineering
National Taiwan Normal University

2. Quiz 4: Question 1 (1/2)
We are told that the joint PDF of random variables X and Y is a constant in the “shaded” area of the figure shown below.
(1) Find (or draw) the PDF of X.
(2) Find (or draw) the PDF of Y.

3. Quiz 4: Question 1 (2/2)
(3) Find the expectation of X . Reference to textbook page 145
(4) Find the variance of X .
Count for variance of X

4. Quiz 4: Question 2 (1/2) Answer: (1-0.6915)*2=0.617 1-0.617=0.383
We are told that is a normal distribution with mean 50 and variance 400.
(1) Find the probability that the value of is in the interval [40 , 60] (given that CDF value of a standard normal is ). Reference to textbook page 157
Answer: ( )*2=0.617
=0.383 40 50 60

5. Quiz 4: Question 2 (2/2) mean= variance= Z is a normal
(2) Find the mean and variance of the random variable Z that has the relation Z=5X+3 . Is Z a normal? Reference to textbook page 154
mean=
variance=
Z is a normal

6. Quiz 4: Question 3 (1/2)
Let X and Y be independent random variables, with each one uniformly distributed in the interval [0, 1]. Find the probability of each of the following events.
(1) (2) x y x y

7. Quiz 4: Question 3 (2/2)
(3) x y 1
The Answer is:
1/5 1

8. Quiz 4: Question 4 Consider a random variable X with PDF
and let A be the event Calculate E[X |A].

9. Quiz 5: Question 1 (1/2)
Given that X is a continuous random variable with PDF and Show that the PDF of random variable Y can be expressed as:
Reference to textbook page 183

10. Quiz 5: Question 1 (2/2)
Chain rule

11. Quiz 4: Question 2 (1/4)
We are told that X and Y are two independent random variables. X is uniformly distributed in the interval [0,2] , while Y is uniformly distributed in the interval [0,1].
Reference to textbook page 188, 164
(1) Find the PDF of
Notice that the interval of two independent random variable forms an area of , and the joint PDF can be viewed as the ‘probability per unit area’. Therefore, the probability of per unit area in the problem is Therefore, after calculating the area constrained by , we need to multiply the size of the area by to obtain the corresponding probability mass. x y x y
x+y

12. while w is in the assigned interval
Quiz 5: Question 2 (2/4)
The answer : x y
Represent the area
while w is in the assigned interval
Multiplied by the probability of unit area

13. Quiz 5: Question 2 (3/4)
(2) Find the PDF of x y

14. Quiz 5: Question 2 (4/4) y The answer: x
Multiplied by the probability of unit area

15. Quiz 5: Question 3 (1/4)
Given that X is an exponential random variable with parameter
Show that the transform (moment generating function) of X can be expressed as:

16. Quiz 5: Question 3 (2/4)
(2) Find the expectation and variance of based on its transform. Reference to textbook page 213 to 215

17. Quiz 5: Question 3 (3/4)
(3) Given that random variable Y can be expressed as Find the transform of Y Reference to textbook page 217

18. Quiz 5: Question 3 (4/4)
(4) Given that Z is also an exponential random variable with parameter , and X and Z are independent. Find the transform of random variable Reference to textbook page 217 to 219