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

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.

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

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 0.6915). Reference to textbook page 157 Answer: (1-0.6915)*2=0.617 1-0.617=0.383 40 50 60

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

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

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

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

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

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

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

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

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

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

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:

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

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

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