A First Course in Probability (10th Edition)
10th Edition
ISBN: 9780134753119
Author: Sheldon Ross
Publisher: PEARSON
expand_more
expand_more
format_list_bulleted
Related questions
Question
![**Title: Probability and Statistics - Random Variables and Expectation**
**Description:**
This section explores the properties of two independent random variables, \( X \) and \( Y \), each uniformly distributed on the interval \([0, 1]\). We examine the random variable \( Z \) which is defined in terms of \( X \) and \( Y \).
**Problem Statement:**
Two independent random variables, \( X \) and \( Y \), are both uniformly distributed on \([0, 1]\). The random variable \( Z \) is defined by
\[
Z = (X - \mu_x)^2 + (Y - \mu_y)^2,
\]
where \( \mu_x = E[X] \) and \( \mu_y = E[Y] \).
(a) Determine the expected value of \( Z \).
(b) Determine the variance of \( Z \).
**Analysis:**
- **Expectation:** The expected value, or mean, of \( Z \) gives insight into the average outcome you can expect from the random variable based on its probability distribution.
- **Variance:** The variance of \( Z \) describes how much the outcomes differ from the expected value, providing a measure of the spread or dispersion in the distribution.
**Note:** This problem involves calculating the expectation and variance for squared deviations of independent random variables. It's an example of integrating basic principles of probability and statistics, such as expectation and variance, with properties of uniform distributions.](https://content.bartleby.com/qna-images/question/af3928cb-73bf-4270-84ca-805354b4b59c/d5332f1c-2a4e-40f4-aa88-a0bd0c89d6ff/6o4djm_processed.jpeg)
Transcribed Image Text:**Title: Probability and Statistics - Random Variables and Expectation**
**Description:**
This section explores the properties of two independent random variables, \( X \) and \( Y \), each uniformly distributed on the interval \([0, 1]\). We examine the random variable \( Z \) which is defined in terms of \( X \) and \( Y \).
**Problem Statement:**
Two independent random variables, \( X \) and \( Y \), are both uniformly distributed on \([0, 1]\). The random variable \( Z \) is defined by
\[
Z = (X - \mu_x)^2 + (Y - \mu_y)^2,
\]
where \( \mu_x = E[X] \) and \( \mu_y = E[Y] \).
(a) Determine the expected value of \( Z \).
(b) Determine the variance of \( Z \).
**Analysis:**
- **Expectation:** The expected value, or mean, of \( Z \) gives insight into the average outcome you can expect from the random variable based on its probability distribution.
- **Variance:** The variance of \( Z \) describes how much the outcomes differ from the expected value, providing a measure of the spread or dispersion in the distribution.
**Note:** This problem involves calculating the expectation and variance for squared deviations of independent random variables. It's an example of integrating basic principles of probability and statistics, such as expectation and variance, with properties of uniform distributions.
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
Trending nowThis is a popular solution!
Step by stepSolved in 3 steps
Knowledge Booster
Similar questions
- Suppose that a random variable X has the following moment generating function, < ¼/7. MX(t) = = (1-9t)−7, t< (a) Find the mean of X. (b) Find the variance of X.arrow_forwardFind E (XY) where X and Y are independent U (0, 1) random variables.arrow_forwardWe have a random variable X and Y that jave the joint pdf f(x) = {1 0<x<1, 0<y<1} {0 otherwise} If U = Y-X2 , what is the support for the random variable U? What would fu(u) and Fu(u) be? Say U = Y/X, what is the support for the random variable U? What would fu(u) and Fu(u) be?arrow_forward
- 1. Consider the Gaussian distribution N (m, σ2).(a) Show that the pdf integrates to 1.(b) Show that the mean is m and the variance is σ.arrow_forwardLet X and Y have joint PDFfX,Y (x, y) = 3x^2y + 3xy^2 if 0 < x < 1, and 0 < y < 1 0 otherwise.a) Find the covariance of X and Y .b) Find Var(X).c) Find the correlation of X and Y (X and Y have the same variance.)arrow_forward
arrow_back_ios
arrow_forward_ios
Recommended textbooks for you
- A First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSON
A First Course in Probability (10th Edition)
Probability
ISBN:9780134753119
Author:Sheldon Ross
Publisher:PEARSON