Read dependence from a joint table

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Problem

Binary variables $X,Y\in\{0,1\}$ have joint probabilities

p_{00}=0.3,\quad p_{01}=0.2,\quad p_{10}=0.1,\quad p_{11}=0.4

Find $P(X=1)$ , $P(Y=1)$ , $P(X=1\mid Y=1)$ , $\operatorname{Cov}(X,Y)$ , and $\operatorname{Corr}(X,Y)$ .

Reveal answer or reference solution

The marginals are $P(X=1)=0.1+0.4=0.5$ and $P(Y=1)=0.2+0.4=0.6$ . Thus $P(X=1\mid Y=1)=0.4/0.6=2/3$ . Since $E[XY]=P(X=Y=1)=0.4$ ,

\operatorname{Cov}(X,Y)=0.4-(0.5)(0.6)=0.1

The Bernoulli variances are $0.25$ and $0.24$ , so

\operatorname{Corr}(X,Y)=\frac{0.1}{\sqrt{0.25\cdot0.24}} =\frac1{\sqrt6}\approx0.4082

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Read dependence from a joint table

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