# Binary variable uncorrelatedness implies independence

I am reading a statistics paper. It said generally uncorrelatedness does not imply independence. But if the case is two binary variables, then the uncorrelatedness can imply independence. How to show this statement?

• Could you link the paper? Aug 5 at 19:56
• "Uncorrelatedness implying independence: Although uncorrelatedness usually does not imply independence, it is well known that it does for two binary variables." Aug 5 at 19:59
• The paper is Zhang, K. (2019). BET on independence Aug 5 at 19:59
• It’s also true for normally distributed random variables
– Bey
Aug 10 at 13:26

Let $$X,Y$$ be two binary random variables with $$P(X=1)=p_X,P(Y=1)=p_Y$$.

Note that zero correlation implies zero covariance and we get:

$$Cov[X,Y] = E[XY]-E[X]E[Y] = 0 \implies E[XY]=E[X]E[Y]=p_Xp_Y$$

For brevity let's define the events $$X_i:= \{X=i\}$$ and $$Y_i:=\{Y=i\}$$

Define $$Z=XY \in \{0,1\}$$

$$E[XY]= P(Z=1) = P(X_1,Y_1)=p_Xp_Y \implies X_1 \perp Y_1$$

In general, $$A\perp B \implies A^c \perp B, \;A \perp B^c, \;A^c \perp B^c$$ (see here) therefore,

$$X_1 \perp Y_1 \implies X_1^c \perp Y_1, \;X_1 \perp Y_1^c, \;X_1^c \perp Y_1^c$$

This happens to exhaust the sample space of the joint experiment $$(X,Y)$$ so we conclude:

$$X_i \perp Y_j\;\;\forall i,j \in \{0,1\} \implies X \perp Y$$