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If one fair six-sided die is rolled, suppose that $X$ is the total number of even numbers shown and $Y$ is the total number of fives shown.

How can I go about calculating the correlation exactly in terms of $p_1$ and $p_2$?

I was given, Hint: Notice that $E(XY) = 0$. Explain why this is true and use this fact in your calculation.

I'm not sure how to do about this... any ideas on where to start?

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up vote 1 down vote accepted

Hints to start:

  • The covariance is $E[(X-E[X])(Y-E[Y])] = E[XY]-E[X]E[Y]$

  • The correlation coefficient is the covariance divided by the product of the standard deviations.

  • For a Bernoulli random variable which is $1$ with probability $p$ and $0$ with probability $1-p$, the expectation is $p$ and the standard deviation is $\sqrt{p(1-p)}$

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okay I calculated E[x] = 1/2, Var(x) = 1/2, E[Y] = 1/6, Var(Y)= sqrt 5/6.. but then I get the Corr(XY) = 0 but the hint says it should be negative.. any idea what I did wrong? – ulc8 Dec 3 '12 at 22:22
@ulc8: The covariance is $0-\frac12 \times \frac16 = -\frac{1}{12}$ so the correlation coefficient will also be negative. And it the standard deviations not the variances which are $\frac12$ and $\sqrt{\frac{5}{36}}$ – Henry Dec 3 '12 at 22:30
thank you.. that makes much more sense! – ulc8 Dec 3 '12 at 23:05

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