We are given a joint probability distribution table for two random variables $X$ and $Y$. Let's extract the values first to determine the marginal distributions, expectations, standard deviations, covariance, and correlation.

1. Joint Probability Distribution Table

Y / X	-2	-1	4	5	Marginal $P_X$
1	0.1	0.2	0	0.3	$P(X = 1) = 0.6$
0	0.1	0.1	0	0	$P(X = 0) = 0.2$
2	0	0	0.2	0	$P(X = 2) = 0.2$

Step 1: Marginal Probability Distributions of X and Y

• Marginal probability distribution of $X$

We sum over all the values of $Y$ for each value of $X$:

$P(X = 1) = 0.1 + 0.2 + 0 + 0.3 = 0.6$ $P(X = 0) = 0.1 + 0.1 + 0 + 0 = 0.2$ $P(X = 2) = 0 + 0 + 0.2 + 0 = 0.2$

Thus, the marginal distribution of $X$ is:

$X$	1	0	2
$P_X$	0.6	0.2	0.2

• Marginal probability distribution of $Y$

We sum over all the values of $X$ for each value of $Y$:

$P(Y = -2) = 0.1 + 0.1 + 0 = 0.2$ $P(Y = -1) = 0.2 + 0.1 + 0 = 0.3$ $P(Y = 4) = 0 + 0 + 0.2 = 0.2$ $P(Y = 5) = 0.3 + 0 + 0 = 0.3$

Thus, the marginal distribution of $Y$ is:

$Y$	-2	-1	4	5
$P_Y$	0.2	0.3	0.2	0.3

Step 2: Expectations of $X$ and $Y$

• Expectation of $X$

$E(X) = \sum_{x} x \cdot P_X(x) = 1 \cdot 0.6 + 0 \cdot 0.2 + 2 \cdot 0.2 = 0.6 + 0.4 = 1$

• Expectation of $Y$

$E(Y) = \sum_{y} y \cdot P_Y(y) = (-2) \cdot 0.2 + (-1) \cdot 0.3 + 4 \cdot 0.2 + 5 \cdot 0.3$ $E(Y) = -0.4 - 0.3 + 0.8 + 1.5 = 1.6$

Step 3: Standard Deviation of $X$ and $Y$

• Variance of $X$

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