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...

See similar textbooks

Related questions

Q: Suppose a continuous random variable X has the following CDF:: F(x) = 1 - 1/ (x+1)4, x > 0. Find SX…

A: We have given, A continuous random variable X has the following CDF: F(x) = 1 - 1x+14, x > 0.

Q: Only 70% of the pints of human blood donated at blood banks are suitable for hospital use. The…

A: Given 70% of the pints of human blood donated at blood banks are suitable for hospital use.…

Q: 2. Suppose that X is a random variable with E(X) = 6 and Var(X) = 4. Find the following: a) Е(4X +…

Q: Let X, Y are random variables (r.v.) and a, b, c, d are values. Complete the formulas using the…

A: Step 1: Step 2: Step 3: Step 4:

Q: iv) E[E[Y²|X]]_ (EY)²

A: Note: As per you request, we have helped you solve only Part (iv) of the question.

Q: 7) Let X₁, X2,...Xn i.i.d random sample from U(0, 0). Consider two estimators of 6₁ = 2X₂ and 6₂ =…

Q: If X is a random variable, prove that Cov(X,X) = σX².

A: Let X is a random variable.

Q: 3. Let the random variable X have the pmf f(x) = = a) E(X) b) E(X²) c) E(3X²2X + 4) (x+1)² for x =…

A: The provided information is as follows:The probability mass function of is .

Q: Suppose that X₁, X2, X3 are independent and identically distributed random variables with…

A: Solution

Q: Each front tire on a particular type of vehicle is supposed to be filled to a pressure of 26 psi.…

A: Understanding the Pressure Relationship in TiresKnowing the air pressure in one tire of a vehicle…

Q: Exercise 8. Suppose X1, X2, , Xn are independent random variables with X; ~ N(Hi, o), 1<i<n. If X =…

Q: Let X1, X2, X3 and X4 be exponential(1) random variables. Find the joint distribution of…

A: To find the joint distribution of the random variables , , and, we can use the Jacobian method.…

Q: Let X1 - b(n1, p) and X2 ~ b(n2, p) be independent random variables. a) Find the m.g.f. of Y = X1+…

A: Given that X1 follows b(n1, p) and X2 follows b(n2, p) and they are independent.

Q: When X is a binomial random variable, then the probability X ≤ r successes can be found using the…

A: Given X is a binomial random variable Number of trials=n=167 Probability of success=p=0.051

Q: The CDF of a random variable X is given the function F(x)= Cx² 3x² + x support of X is x=1,2,3,....…

A: Given that, the CDF of a random variable X is given the function F(x)=cx23x2+x…

Q: 2. Let Y₁,..., Yp be independent random variables such that Y~ N(0, 1). Write Y = (Y₁...., Yp) and 0…

A: To show that an unbiased estimator of the risk of the James-Stein estimator iswe can use the…

Q: When X is a binomial random variable, then the probability X ≤ r successes can be found using the…

A: Given X is a binomial random variable n=No. of trials=125 p=Probability of success=0.016

Q: Suppose that X₁, X2, X3 are independent and identically distributed random variables with…

A: Given Cdf F(x)=1-2-x; x≥0

Q: Assume that the random variables X1 and X2 are bivariate normally distributed with mean µ and…

Q: Suppose X₁, X2, · ·, Xn are independent random variables and have CDF F(x). Find the CDF of Y =…

A: To find the cumulative distribution function (CDF) of Y=min{X1,X2,…,Xn}, we need to consider the…

Q: Suppose that X and Y are random variables with E(X) = 2 , E(Y) = 5 and E(X²) = 8, E(Y²) = 30 and…

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

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

This is a popular solution!

SEE SOLUTION Check out a sample Q&A here

Method

VIEW

Solution

VIEW

Trending now

This is a popular solution!

Step by step

Solved in 2 steps with 1 images

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Multivariate Distributions and Functions of Random Variables$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, probability and related others by exploring similar questions and additional content below.

Similar questions

Let X1 and X2 be independent Poisson random variables with respective means lambda1 = 2 and lambda2 = 3. Find (a) P(X1=3,X2=5). (b) P(X1 + X2 = 1).
Suppose we have three independent random variables X, Y and Z where... Var(X + 2Y) = 13, Var(2Y + 4Z) = 40, and Var(Z) = 2. a) If E(Y) = E(Z) = 1, what is E(Y2+Z2)? b) What is SD(X + Y + Z)?
4. Suppose we have random variables W and Q, and we know Var(W) = 25, Var(Q) = 10, and Cov(W, Q) = 2. Now, let A = 2W + 4Q. What is Var(A)?
I am having a tought time with this question
Let (X1, X2, X3) be the random outcome of tossing a fair six-faced die three times, and let (Y(1), Y(2), Y(3) be the order statistics from (X1, X2, X3). We have Y(1) = min (X1, X2, X3), Y(3) = max (X1, X2, X3), and Y(1) < Y(2) < Y(3). Find Pr (Y(1) = 3).
Let (Ω, Pr) be a probability space, and let X and Y be two independent random variables that are positive and have non-zero variance. (a) Prove that X^2 and Y are independent. Note that by symmetry, it will also follow that Y^2 and X are independent. (b) Use the result from part (a) to show that the random variables W = X + Y and Z = XY are positively correlated (i.e. Cov(W, Z) > 0).
Suppose we choose arbitrarily a point from the square with corners at(2,1), (3,1), (2,2), and (3,2). The random variable A is the area of the trianglewith its corners at (2,1), (3,1), and the chosen point. Compute E[A].