Question

110.s90.04 let \\(x\\) and \\(y\\) be discrete random variables with joint probability distribution given below:

\\(\

$$\begin{array}{|l|c|c|c|c|c|c|} \\hline x & -1 & -1 & 0 & 0 & 1 & 1 \\\\ \\hline y & 0 & 1 & 0 & 1 & 0 & 1 \\\\ \\hline p(x,y) & 0.1 & 0.1 & 0.1 & 0.3 & 0.2 & 0.2 \\\\ \\hline \\end{array}$$

\\)

what is \\(\text{cov}(x, y)\\)?

Explanation:

State the covariance formula

We need to find \(\text{Cov}(X, Y)\).
The formula is:

$$ \text{Cov}(X, Y) = E[XY] - E[X]E[Y] $$

Calculate the expected value of X

We find the marginal distribution of \(X\):

$$
LATEXBLOCK0
$$

Now, calculate \(E[X]\):

$$ E[X] = (-1)(0.2) + (0)(0.4) + (1)(0.4) = 0.2 $$

Calculate the expected value of Y

We find the marginal distribution of \(Y\):

$$
LATEXBLOCK1
$$

Now, calculate \(E[Y]\):

$$ E[Y] = (0)(0.4) + (1)(0.6) = 0.6 $$

Calculate the expected value of XY

We compute \(X \cdot Y\) for each pair:

$$
LATEXBLOCK2
$$

Compute the final covariance

Substitute the expected values into the formula:

$$
LATEXBLOCK3
$$

Answer:

\(\text{Cov}(X, Y) = -0.02\)