# Thread: Let random variate X take on a value 1 an 0...show that X and Y are independent

1. ## Let random variate X take on a value 1 an 0...show that X and Y are independent

Question :
Let random variate X take on a values 1 and 0 with the probabilities $\displaystyle p_1$ and $\displaystyle q_1 = 1 - p_1$ respectively and let Y take the values 1 and 0 with probabilities $\displaystyle p_2$ and $\displaystyle q_2$ respectively . If the correlation coefficient between X and Y is 0, show that X and Y are independent.

2. If the correlation, hence the covariance is zero we have

$\displaystyle E(XY)=E(X)E(Y)=p_1p_2$

since $\displaystyle E(X)=p_1$ and $\displaystyle E(Y)=p_2$

That means $\displaystyle P(X=1,Y=1)=p_1p_2=P(X=1)P(Y=1)$

Now you need to fill in the other 3 spots of the 2 by 2 joint probability distribution.

That is...$\displaystyle P(X=1,Y=0)=P(X=1)P(Y=0)$

$\displaystyle P(X=0,Y=1)=P(X=0)P(Y=1)$ and $\displaystyle P(X=0,Y=0)=P(X=0)P(Y=0)$.

and this part doesn't make sense....'show that X and Y is 0'

3. Originally Posted by matheagle
If the correlation, hence the covariance is zero we have

$\displaystyle E(XY)=E(X)E(Y)=p_1p_2$

since $\displaystyle E(X)=p_1$ and $\displaystyle E(Y)=p_2$

That means $\displaystyle P(X=1,Y=1)=p_1p_2=P(X=1)P(Y=1)$

Now you need to fill in the other 3 spots of the 2 by 2 joint probability distribution.

That is...$\displaystyle P(X=1,Y=0)=P(X=1)P(Y=0)$

$\displaystyle P(X=0,Y=1)=P(X=0)P(Y=1)$ and $\displaystyle P(X=0,Y=0)=P(X=0)P(Y=0)$.

and this part doesn't make sense....'show that X and Y is 0'

It is 'show that x and y are independent '

I couldnt get u mite ............where are u getting at.....