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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).
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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
40 50 60 Quiz 4: Question 2 (1/2) • 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
Quiz 4: Question 2 (2/2) • (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
y y x x 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)
y x Quiz 4: Question 3 (2/2) • (3) 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
y y x x 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
Represent the area while w is in the assigned interval y x Quiz 5: Question 2 (2/4) The answer : Multiplied by the probability of unit area
y x Quiz 5: Question 2 (3/4) • (2) Find the PDF of
y x Quiz 5: Question 2 (4/4) The answer: 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