**Example Problem on Expected Value and Variance of Linear Combinations of Independent Random Variables****Problem Statement:**Suppose that \(X_1, X_2,\) and \(X_3\) are independent random variables. Suppose further that\[ E(X_1) = 4 \]\[ E(X_2) = 5 \]\[ E(X_3) = 7 \]and\[ Var(X_1) = 4 \]\[ Var(X_2) = \frac{1}{2} \]\[ Var(X_3) = \sqrt{2} \](**Tasks**)a.) Compute \( E(4X_1 + \pi X_2 + e X_3 + 13) \)b.) Compute \( E(X_1 X_3 + X_2) \)c.) Compute \( Var(\sqrt{2} X_2 + \sqrt{3} X_3 + 17) \)### Solutions:**a.) Compute \( E(4X_1 + \pi X_2 + e X_3 + 13) \)**To find the expectation of a linear combination of random variables, we use the linearity property of expectation:\[ E(aX + bY + c) = aE(X) + bE(Y) + c \]Thus for the given expression:\[ E(4X_1 + \pi X_2 + e X_3 + 13) = 4E(X_1) + \pi E(X_2) + e E(X_3) + 13 \]\[ E(4X_1 + \pi X_2 + e X_3 + 13) = 4(4) + \pi(5) + e(7) + 13 \]\[ E(4X_1 + \pi X_2 + e X_3 + 13) = 16 + 5\pi + 7e + 13 \]\[ E(4X_1 + \pi X_2 + e X_3 + 13) = 29 + 5\pi + 7e \]**b.) Compute \( E(X_1 X_3 + X_2) \)**For the second expectation, we must note that expectation of the product of independent random variables is the product of expectations:

Name: **Example Problem on Expected Value and Variance of Linear Combinatio
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Description: **Example Problem on Expected Value and Variance of Linear Combinations of Independent Random Variables****Problem Statement:**Suppose that \(X_1, X_2,\) and \(X_3\) are independent random variables. Suppose further that\[ E(X_1) = 4 \]\[ E(X_2) = 5

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Suppose that X1, X2, and X3 are independent random variables. Suppose further that E(X1) = 4 E(X2) = 5 E(X3) = 7 and Var(X1) = 4 Var(X2) Var(X3) = v2 a.) Compute E(4X1 +«X2 + eX3+13) b.) Compute E(X1X3 + X2) c.) Compute Var(V2X2+ V3X3+17)

Suppose that X1, X2, and X3 are independent random variables. Suppose further that E(X1) = 4 E(X2) = 5 E(X3) = 7 and Var(X1) = 4 Var(X2) Var(X3) = v2 a.) Compute E(4X1 +«X2 + eX3+13) b.) Compute E(X1X3 + X2) c.) Compute Var(V2X2+ V3X3+17)

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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