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7. If the r.v.'s X and Y have bivariate normal distribution with parameters ₁ = 3.2, 2 = 1.44, 0₁ = 12,02 = 16 and p = 0.7, determine the following quantities: (a) E(X), E(Y), Var(X), Var(Y), p(X, Y), and Cov(X,Y). (b) the distribution of X and the distribution of Y. (c) the probabilities P(0.8 < X < 4.2), P(Y > 14).

7. If the r.v.'s X and Y have bivariate normal distribution with parameters ₁ = 3.2, 2 = 1.44, 0₁ = 12,02 = 16 and p = 0.7, determine the following quantities: (a) E(X), E(Y), Var(X), Var(Y), p(X, Y), and Cov(X,Y). (b) the distribution of X and the distribution of Y. (c) the probabilities P(0.8 < X < 4.2), P(Y > 14).

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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**Problem 7: Bivariate Normal Distribution** Given that the random variables \( X \) and \( Y \) have a bivariate normal distribution with the following parameters: - Mean of \( X \) (\( \mu_1 \)) = 3.2 - Mean of \( Y \) (\( \mu_2 \)) = 1.44 - Standard deviation of \( X \) (\( \sigma_1 \)) = 12 - Standard deviation of \( Y \) (\( \sigma_2 \)) = 16 - Correlation (\( \rho \)) = 0.7 **Determine the following quantities:** (a) \( E(X) \), \( E(Y) \), \( Var(X) \), \( Var(Y) \), \( \rho(X,Y) \), and \( Cov(X,Y) \). (b) The distribution of \( X \) and the distribution of \( Y \). (c) The probabilities \( P(0.8 < X < 4.2) \) and \( P(Y > 14) \). (d) The conditional distribution of \( Y \), given \( X = 3.8 \). (e) The conditional distribution of \( X \), given \( X = 10 \). (f) The probabilities \( P(X > 3.2|Y = 10) \) and \( P(Y < 12|X = 3.8) \). (g) \( E(X|Y = 10) \), \( E(Y|X = 3.8) \), \( Var(X|Y = 10) \), \( Var(Y|X = 3.8) \).
Transcribed Image Text:**Problem 7: Bivariate Normal Distribution** Given that the random variables \( X \) and \( Y \) have a bivariate normal distribution with the following parameters: - Mean of \( X \) (\( \mu_1 \)) = 3.2 - Mean of \( Y \) (\( \mu_2 \)) = 1.44 - Standard deviation of \( X \) (\( \sigma_1 \)) = 12 - Standard deviation of \( Y \) (\( \sigma_2 \)) = 16 - Correlation (\( \rho \)) = 0.7 **Determine the following quantities:** (a) \( E(X) \), \( E(Y) \), \( Var(X) \), \( Var(Y) \), \( \rho(X,Y) \), and \( Cov(X,Y) \). (b) The distribution of \( X \) and the distribution of \( Y \). (c) The probabilities \( P(0.8 < X < 4.2) \) and \( P(Y > 14) \). (d) The conditional distribution of \( Y \), given \( X = 3.8 \). (e) The conditional distribution of \( X \), given \( X = 10 \). (f) The probabilities \( P(X > 3.2|Y = 10) \) and \( P(Y < 12|X = 3.8) \). (g) \( E(X|Y = 10) \), \( E(Y|X = 3.8) \), \( Var(X|Y = 10) \), \( Var(Y|X = 3.8) \).
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