A) You have two fair dice, one with three sides labeled 0,1,2 and one with 4 sides, labeled 0, 1, 2, 3 Let X 1 be the outcome of rolling the first die, and X 2 the outcome of rolling the second The rolls are independent a What is the joint distribution of (X 1 , X 2 ) b Let Y 1 X 1 X 2 and Y 2 max X 1 , X 2 Make a table for the joint distribution ofY 1 , Y 2 c What are the marginal distribution of Y 1 , Y 2 Are they independent B) Use the first part below to solve the second part Let X 1 , , X n be i i d continuous random variables Find the pdf of Y 1 max X 1 , , X n and Y 2 min X 1 , , X n

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Question: A) You have two fair dice, one with three sides labeled 0,1,2 and one with 4 sides, labeled 0, 1, 2, 3. Let X 1

A) You have two fair dice, one with three sides labeled 0,1,2 and one with 4 sides, labeled 0, 1, 2, 3. Let X₁ be the outcome of rolling the first die, and X₂ the outcome of rolling the second. The rolls are independent. a. What is the joint distribution of (X₁, X₂)? b. Let Y₁ = X₁ X₂ and Y₂ = max{X₁, X₂}. Make a table for the joint distribution ofY₁, Y₂. c. What are the marginal distribution of Y₁, Y₂? Are they independent?

B) Use the first part below to solve the second part. Let X₁, . . . , X_n be i.i.d. continuous random variables. Find the pdf of Y₁ = max{X₁, . . . , X_n} and Y₂ = min{X₁, . . . , X_n}.

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