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Introduction to Probability: Solutions for Quizzes 4 and 5

This resource provides solutions for quiz questions related to probability theory, joint PDF, random variables, mean, variance, distribution, and transformations. It covers topics like uniform distribution, normal distribution, and exponential random variables.

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Introduction to Probability: Solutions for Quizzes 4 and 5

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  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. 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

  5. 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

  6. 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)

  7. y x Quiz 4: Question 3 (2/2) • (3) 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. 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

  12. 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

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

  14. y x Quiz 5: Question 2 (4/4) The answer: 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

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