Let's consider three independent random variables $X_1$, $X_2$, and $X_3$. The expected values and variances of these random variables are given as follows:- $E(X_1) = \frac{7}{8}$- $E(X_2) = \frac{5}{9}$- $E(X_3) = \frac{4}{7}$Further, the variances are given by:- $Var(X_1) = 5$- $Var(X_2) = 10$- $Var(X_3) = 15$We are tasked with computing two quantities:1. $E(\pi X_1 + 4X_2 + \sqrt{3}X_3 + 15)$2. $Var(\sqrt{2}X_1 + \sqrt{3}X_2 + 7X_3 + \frac{4}{5})$### Solutions:1. **Expected Value Calculation:** To compute $E(\pi X_1 + 4X_2 + \sqrt{3}X_3 + 15)$, we use the linearity property of expectation: $E(\pi X_1 + 4X_2 + \sqrt{3}X_3 + 15) = \pi E(X_1) + 4E(X_2) + \sqrt{3}E(X_3) + E(15)$ Since the expected value of a constant is the constant itself, $E(15) = 15$: So, \[ E(\pi X_1 + 4X_2 + \sqrt{3}X_3 + 15) = \pi \cdot \frac{7}{8} + 4 \cdot \frac{5}{9} + \sqrt{3} \cdot \frac{4}{7} + 15 \]2. **Variance Calculation:** To compute $Var(\sqrt{2}X_1 + \sqrt{3}X_2 + 7X_3 + \frac{4}{5})$, we use the fact that the variance of a constant is zero and that for independent random variables $X$ and $Y$, \(Var(aX + bY

Using linear expansion of expectations ans variance

Let X1, X2, and X3 be independent random variables such that E(X1) ‚E(X2) and E(X3) Further, suppose that Var(X1) = 5, Var(X2) = 10, and Var(X3) = 15. 1.) Compute E(TX1 + 4X2 + V3X3+ 15) 2.) Compute Var(/2X1 + V3X2 + 7X3+

Let X1, X2, and X3 be independent random variables such that E(X1) ‚E(X2) and E(X3) Further, suppose that Var(X1) = 5, Var(X2) = 10, and Var(X3) = 15. 1.) Compute E(TX1 + 4X2 + V3X3+ 15) 2.) Compute Var(/2X1 + V3X2 + 7X3+

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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Question

Let's consider three independent random variables $X_1$, $X_2$, and $X_3$. The expected values and variances of these random variables are given as follows:

- $E(X_1) = \frac{7}{8}$
- $E(X_2) = \frac{5}{9}$
- $E(X_3) = \frac{4}{7}$

Further, the variances are given by:
- $Var(X_1) = 5$
- $Var(X_2) = 10$
- $Var(X_3) = 15$

We are tasked with computing two quantities:

1. $E(\pi X_1 + 4X_2 + \sqrt{3}X_3 + 15)$

2. $Var(\sqrt{2}X_1 + \sqrt{3}X_2 + 7X_3 + \frac{4}{5})$

### Solutions:

1. **Expected Value Calculation:**

To compute $E(\pi X_1 + 4X_2 + \sqrt{3}X_3 + 15)$, we use the linearity property of expectation:

$E(\pi X_1 + 4X_2 + \sqrt{3}X_3 + 15) = \pi E(X_1) + 4E(X_2) + \sqrt{3}E(X_3) + E(15)$

Since the expected value of a constant is the constant itself, $E(15) = 15$:

So,
\[
E(\pi X_1 + 4X_2 + \sqrt{3}X_3 + 15) = \pi \cdot \frac{7}{8} + 4 \cdot \frac{5}{9} + \sqrt{3} \cdot \frac{4}{7} + 15
\]

2. **Variance Calculation:**

To compute $Var(\sqrt{2}X_1 + \sqrt{3}X_2 + 7X_3 + \frac{4}{5})$, we use the fact that the variance of a constant is zero and that for independent random variables $X$ and $Y$, \(Var(aX + bY

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