the correlation between two random variables X and Y is p=-0.75.it is given that X=1,2or3 with equal probability of 1/3 and Y=-1,-2or-3 with equal probability of 1/3.the covariance is equal to?i need help.i have been struggling to get the answer>
To find the covariance between two random variables X and Y, we can use the formula:
Cov(X, Y) = E[(X - E[X])(Y - E[Y])]
First, let's find E[X] and E[Y], which are the expected values of X and Y, respectively.
E[X] = (1 * 1/3) + (2 * 1/3) + (3 * 1/3) = 2
E[Y] = (-1 * 1/3) + (-2 * 1/3) + (-3 * 1/3) = -2
Next, we need to calculate E[(X - E[X])(Y - E[Y])]. Since we know the correlation between X and Y (p = -0.75), we can find this by using the formula:
Cov(X, Y) = p * σ(X) * σ(Y)
where σ(X) and σ(Y) are the standard deviations of X and Y, respectively.
The standard deviation of X can be calculated as follows:
σ(X) = √ Var(X)
Var(X) = E[(X - E[X])^2]
Var(X) = [(1-2)^2 * 1/3] + [(2-2)^2 * 1/3] + [(3-2)^2 * 1/3] = 2/3
σ(X) = √(2/3) ≈ 0.82
Similarly, the standard deviation of Y is:
σ(Y) = √ Var(Y)
Var(Y) = E[(Y - E[Y])^2]
Var(Y) = [(-1+2)^2 * 1/3] + [(-2+2)^2 * 1/3] + [(-3+2)^2 * 1/3] = 2/3
σ(Y) = √(2/3) ≈ 0.82
Now, let's substitute the values into the formula:
Cov(X, Y) = -0.75 * 0.82 * 0.82
Cov(X, Y) ≈ -0.492
Therefore, the covariance between X and Y is approximately -0.492.