**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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A First Course in Probability (10th Edition)

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Chapter1: Combinatorial Analysis

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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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