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![**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)
