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A coin (with probability of getting head equal to $p$) is tossed twice. Let $X$ be the total number of heads and $Y$ be the difference between the total number of heads and the total number of tails. Find $\rho(X, Y)$.

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    Three comments by the OP now deleted by the OP.2012-12-16

1 Answers 1

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If the coin is tossed twice, there are four possible outcomes, corresponding to the following values of $X$ and $Y$. (I take the definition of $Y$ to mean "number of heads (H) minus number of tails (T)")

event  probability   X  Y HH     pp            2  2 HT     p(1-p)        1  0 TH     (1-p)p        1  0 HH     (1-p)(1-p)    0  -2 

To compute the correlation between $X$ and $Y$ (I assume that's what you mean by $\rho(X,Y)$), we observe that $E X = 2p^2+2p(1-p)=2p$, $E X^2 = 4p^2+2p(1-p)=2p(p+1)$ and thus $ \operatorname{Var} X = E X^2 - (E X)^2 = 2p(1-p). $ Similarly, one computes $E Y=2p^2-2(1-p)^2=2(2p-1)$, $E Y^2=4p^2+4(1-p)^2=8p^2-8p+4$, and thus $ \operatorname{Var} Y = E Y^2 - (E Y)^2 = -8p^2+8p=8p(1-p). $ One also computes that $EXY=4p^2$, and thus the covariance between $X$ and $Y$ is $ \operatorname{Cov}(X,Y) =E XY-E X EY = 4p^2 - 16p^2(1-p)^2. $

Finally, the covariance between $X$ and $Y$ becomes $ \rho(X,Y)=\frac{\operatorname{Cov}(X,Y)}{\sqrt{\operatorname{Var} X \operatorname{Var} Y}} = \frac{p \left(4p(2-p)-3\right)}{1-p}. $

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    Eckhard This is a friendly advice: when the OP forgets (or refuses) to show what they tried or what they know, one may prefer to provide hints rather than a full solution (I know, this is not always easy). To get the idea, you might wish to consult answers to other questions by same OP.2012-12-16