Independent random variables are uncorrelated. In general, uncorrelated random variables need not be independent. But in specific examples they are.

Question

Question: Let ξ and η be Bernoulli(p) and Bernoulli(r) random variables, 0 < p, r < 1. Show that if ξ and η are uncorrelated then they are independent.

My attempt: I'm not exactly sure where to begin with this. I know that if the covariance between two random variables is equal to zero, then they are said to be uncorrelated. However, I struggled to complete this.

score 2 · Answer 1 · answered Apr 20 '18 at 14:35

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To show that two discrete variables are independent, it suffices to show that $P(X=a\cap Y=b)=P(X=a)P(Y=b)$ for all values $a,b$ that $X,Y$ can take. Here, $a,b$ can both only be $0$ and $1$. I'll take care of the case $a=b=1$.

The correlation between $\xi$ and $\eta$ is equal to $\frac{E[\xi \eta]-E[\xi]E[\eta]}{\sqrt{\text{Var }\xi\text{ Var }\eta}}$. If this is zero, then this implies $$ E[\xi\eta]=E[\xi]E[\eta]=p\cdot r=P(\xi=1)P(\eta=1) $$

To find $E[\xi\eta]$, note that $\xi\eta$ is always equal to $0$ or $1$, so $$ E[\xi\eta] = P(\xi\eta=1)=P(\xi=1\cap \eta=1) $$

The above proves that $$P(\xi=1\cap \eta=1)=P(\xi=1)P(\eta=1).$$

answered Apr 20 '18 at 14:35

Mike Earnest

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Thanks for your answer Mike! Just wondering how did you get the correlation between ξ and η? – user543776 Apr 20 '18 at 14:38
Correlation is covariance over the product of the standard deviations. The numerator is the usual shortcut formula for calculating covariance https://en.wikipedia.org/wiki/Covariance#Definition – Mike Earnest Apr 20 '18 at 14:41
Got you! Thanks so much for your help! – user543776 Apr 20 '18 at 14:42

Independent random variables are uncorrelated. In general, uncorrelated random variables need not be independent. But in specific examples they are.

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