a. Explanation Calculation: It is given that the joint density function of the random variable X and Y is f ( x , y ) = c 1 − x 2 − y 2 ; x 2 + y 2 ≤ 1 and the inequality explains a unit circle centered at the origin ( 0 , 0 ) . If x is fixed, then y 2 ≤ 1 − x 2 ⇒ y ∈ [ − 1 − x 2 , 1 − x 2 ] . This implies that the integration limits for integrating over the values of y is y ∈ [ − 1 − x 2 , 1 − x 2 ] . Therefore, 1 = ∫ − 1 1 ∫ − 1 − x 2 1 − x 2 f ( x , y ) d y d x ( x ∈ [ − 1 , 1 ] y ∈ [ − 1 − x 2 , 1 − x 2 ] ) = ∫ − 1 1 ( ∫ − 1 − x 2 1 − x 2 c 1 − x 2 − y 2 d y ) d x = c ∫ − 1 1 1 − x 2 ( ∫ − 1 − x 2 1 − x 2 1 − y 2 1 − x 2 d y ) d x = | y 1 − x 2 = sin t y = − 1 − x 2 ⇒ sin t = − 1 ⇒ t = − π 2 d y = 1 − x 2 cos t d t y = − 1 − x 2 ⇒ sin t = 1 ⇒ t = π 2 | Further simplification, 1 = c ∫ − 1 1 1 − x 2 ( ∫ − π 2 π 2 1 − sin 2 t cos t d t ) d x = c ∫ − 1 1 1 − x 2 ( ∫ − π 2 π 2 1 − sin 2 t ︸ cos < b. To determine Sketch the joint density. c. To determine Find P ( X 2 + Y 2 ) ≤ 1 2 . d. To determine Find the marginal densities of X and Y and also check whether the random variables are independent or not. e. To determine Find the conditional densities.

Mathematical Statistics and Data Analysis

3rd Edition

ISBN: 9781111793715

Author: John A. Rice

Publisher: Cengage Learning

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Question

Chapter 3.8, Problem 15P

To determine

Find c.

To determine

Sketch the joint density.

To determine

Find P(X2+Y2)≤12.

To determine

Find the marginal densities of X and Y and also check whether the random variables are independent or not.

To determine

Find the conditional densities.

Expert Solution & Answer

Chapter 3 Solutions

Mathematical Statistics and Data Analysis

Ch. 3.8 - Prob. 1P Ch. 3.8 - Prob. 2P Ch. 3.8 - Prob. 3P Ch. 3.8 - Prob. 4P Ch. 3.8 - Prob. 5P Ch. 3.8 - Prob. 6P Ch. 3.8 - Prob. 7P Ch. 3.8 - Prob. 8P Ch. 3.8 - Prob. 9P Ch. 3.8 - Prob. 10P

Ch. 3.8 - Prob. 11P Ch. 3.8 - Prob. 12P Ch. 3.8 - Prob. 13P Ch. 3.8 - Prob. 14P Ch. 3.8 - Prob. 15P Ch. 3.8 - Prob. 16P Ch. 3.8 - Prob. 17P Ch. 3.8 - Prob. 18P Ch. 3.8 - Prob. 19P Ch. 3.8 - Prob. 20P Ch. 3.8 - Prob. 22P Ch. 3.8 - Prob. 23P Ch. 3.8 - Prob. 24P Ch. 3.8 - Prob. 25P Ch. 3.8 - Prob. 27P Ch. 3.8 - Prob. 28P Ch. 3.8 - Prob. 29P Ch. 3.8 - Prob. 30P Ch. 3.8 - Prob. 31P Ch. 3.8 - Prob. 32P Ch. 3.8 - Prob. 33P Ch. 3.8 - Prob. 34P Ch. 3.8 - Prob. 35P Ch. 3.8 - Prob. 38P Ch. 3.8 - Prob. 39P Ch. 3.8 - Prob. 44P Ch. 3.8 - Prob. 45P Ch. 3.8 - Prob. 46P Ch. 3.8 - Prob. 47P Ch. 3.8 - Prob. 48P Ch. 3.8 - Prob. 50P Ch. 3.8 - Prob. 51P Ch. 3.8 - Prob. 52P Ch. 3.8 - Prob. 53P Ch. 3.8 - Prob. 54P Ch. 3.8 - Prob. 55P Ch. 3.8 - Prob. 56P Ch. 3.8 - Prob. 57P Ch. 3.8 - Prob. 58P Ch. 3.8 - Prob. 60P Ch. 3.8 - Prob. 61P Ch. 3.8 - Prob. 62P Ch. 3.8 - Prob. 63P Ch. 3.8 - Prob. 64P Ch. 3.8 - Prob. 65P Ch. 3.8 - Prob. 66P Ch. 3.8 - Prob. 67P Ch. 3.8 - Prob. 68P Ch. 3.8 - Prob. 69P Ch. 3.8 - Prob. 70P Ch. 3.8 - Prob. 71P Ch. 3.8 - Prob. 72P Ch. 3.8 - Prob. 73P Ch. 3.8 - Prob. 74P Ch. 3.8 - Prob. 75P Ch. 3.8 - Prob. 76P Ch. 3.8 - Prob. 77P Ch. 3.8 - Prob. 78P Ch. 3.8 - Prob. 79P Ch. 3.8 - Prob. 80P Ch. 3.8 - Prob. 81P

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