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The image you uploaded contains a problem set related to joint probability distributions of two random variables \(X\) and \(Y\), with a probability density function \( f_{XY}(x, y) \) defined as:

\[
f_{XY}(x, y) = 
\begin{cases} 
c & \text{if } (x, y) \in S_{XY} \\
0 & \text{otherwise} 
\end{cases}
\]

where \( S_{XY} \) is the region of a triangle with vertices at points \( (-1,0) \), \( (1,0) \), and \( (0,1) \), and \( c \) is a constant.

The questions are as follows:

1. **(a)** Calculate the value of the constant \( c \).
2. **(b)** Compute the probability \( P(Y < X) \).
3. **(c)** Compute the probability \( P(Y \leq 1) \).
4. **(d)** Examine whether \( X \) and \( Y \) are independent.
5. **(e)** Determine the marginal probability density functions of \( X \) and \( Y \).
6. **(στ)** Compute the expected values \( E(X) \) and \( E(Y) \).
7. **(ζ)** Determine the set of \( x \) values for which the conditional probability distribution \( f_{Y|X}(y|x) \) is defined. Then, find \( f_{Y|X}(y|x) \) for each \( x \).
8. **(η)** Compute the covariance \( \text{Cov}(X, Y) \).

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The image contains a problem set related to the joint probability distribution of two random variables X and Y, with a probability density function f_{XY}(x, y) defined as a constant c over a triangular region with vertices at (-1, 0), (1, 0), and (0, 1). The set of problems includes calculating probabilities, checking independence, and determining marginal distributions and covariances.

Solve problems related to the joint probability distribution of two random variables X and Y, including calculating probabilities, marginal distributions, expected values, and covariance. The triangular region is defined by vertices at (-1, 0), (1, 0), and (0, 1), with key concepts in probability theory explored.

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