An "independence challenge", perhaps: Let X and Y be independent RVs, where X is uniform in [−1, 1] and Y is discrete and has PMF: Py(y) = = { 1-p_y=-1 y = 1 otherwise Р 0 Consider new RV, Z = XY. For each of the following statements find if it is true or false: i) Y and Z are independent; and ii) X and Z are independent
An "independence challenge", perhaps: Let X and Y be independent RVs, where X is uniform in [−1, 1] and Y is discrete and has PMF: Py(y) = = { 1-p_y=-1 y = 1 otherwise Р 0 Consider new RV, Z = XY. For each of the following statements find if it is true or false: i) Y and Z are independent; and ii) X and Z are independent
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![**Independence Challenge**
Consider an "independence challenge," where we have the following setup:
1. Let \(X\) and \(Y\) be independent random variables (RVs).
2. \(X\) is uniformly distributed in the interval \([-1, 1]\).
3. \(Y\) is discrete and has the probability mass function (PMF):
\[
P_Y(y) =
\begin{cases}
1 - p & \text{if } y = -1 \\
p & \text{if } y = 1 \\
0 & \text{otherwise}
\end{cases}
\]
### Problem Statement:
Consider a new random variable \(Z\) defined as \(Z = XY\).
Determine the truth or falsehood of the following statements:
i) \(Y\) and \(Z\) are independent.
ii) \(X\) and \(Z\) are independent.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F099a9854-7cc3-46b9-a37c-ec0486b4a12a%2F066fe08f-da73-4689-9fa0-6427440d3b75%2Fxh78yru_processed.png&w=3840&q=75)
Transcribed Image Text:**Independence Challenge**
Consider an "independence challenge," where we have the following setup:
1. Let \(X\) and \(Y\) be independent random variables (RVs).
2. \(X\) is uniformly distributed in the interval \([-1, 1]\).
3. \(Y\) is discrete and has the probability mass function (PMF):
\[
P_Y(y) =
\begin{cases}
1 - p & \text{if } y = -1 \\
p & \text{if } y = 1 \\
0 & \text{otherwise}
\end{cases}
\]
### Problem Statement:
Consider a new random variable \(Z\) defined as \(Z = XY\).
Determine the truth or falsehood of the following statements:
i) \(Y\) and \(Z\) are independent.
ii) \(X\) and \(Z\) are independent.
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