Covariance of a discrete distribution

Probability Level 3

Let $X$ and $Y$ be random variables such that

$P(X = 0, Y = -1) = 1/5$
$P(X = 0, Y = 1) = 1/5$
$P(X = 1, Y = -1) = 1/2$
$P(X = 1, Y = 1) = 1/10$ .

Find $\text{Cov}(X, Y)$ .

The answer is -0.16.

1 solution

Lawrence Chiou
Apr 7, 2016

We first compute

$P(XY = -1) = 1/2$ , $P(XY = 0) = 2/5$ , and $P(XY = 1) = 1/10$ , so $E[XY] = -2/5$ .
$P(X = 0) = 2/5$ and $P(X = 1) = 3/5$ , so $E[X] = 3/5$ .
$P(Y = -1) = 7/10$ and $P(Y = 1) = 3/10$ , so $E[Y] = -2/5$ .

It follows that

$\text{Cov}(X, Y) = E[XY] - E[X] E[Y] = -2/5 - (3/5)(-2/5) = \boxed{-4/25.}$

Is there a way we could intuitively have known that the covariance was going to be negative from looking at the joint distribution given?

Eli Ross Staff - 5 years, 2 months ago

X being zero has no effect on covariance since Y is equally likely to be -1 or 1. Then, just looking at P(X=1) you can see Y is more likely to be negative with respect to X.

M B - 3 years ago

E[XY] is wrong. E[XY] = -1/5 since it is E[XY] = 0 -1 1/5 + 0 1 1/5+1 (-1) 1/2+1 1 1/10 = -1/5

Christian Willdoner - 7 months, 3 weeks ago

It looks right to me. E[XY] = (0)(1/5) + (0)(1/5) + (-1)(1/2) + (1)(1/10) = 0 - 1/2 + 1/10 = -2/5

Tony Bai - 4 months, 1 week ago

Covariance of a discrete distribution

The answer is -0.16.

1 solution

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