Question

Let X and Y be two Bernoulli distributed variables. Furthermore, P(X=0; Y=0)=0:3, P(X=0; Y=1)=0:2 and P(X=1; Y=0)=0:2. Which of the following statements is correct? (a). X and Y are correlated and independent (b). X and Y are negatively correlated and dependent (c). X and Y are uncorrelated and independent (d). X and Y are positively correlated and dependent

Answer 1

$P(X=0)=$

$P(X=0,Y=0)+P(X=0,Y=1)=$

$0.3+0.2=0.5$,

$E(X)=0.5\cdot0+0.5\cdot1=0.5$,

$P(Y=0)=$

$P(X=0,Y=0)+P(X=1,Y=0)=$

$0.3+0.2=0.5$,

$E(Y)=0.5\cdot0+0.5\cdot1=0.5$,

$cov(X,Y)=$

$0.3(0-0.5)(0-0.5)+$

$0.2(0-0.5)(1-0.5)+$

$0.2(1-0.5)(0-0.5)+$

$(1-0.3-0.2-02)(1-0.5)(1-0.5)=$

$0.3(-0.5)(-0.5)+0.2(-0.5)0.5+$

$0.2\cdot0.5(-0.5)+0.3\cdot0.5\cdot0.5=0.05>0,\implies$

$corr(X,Y)>0$,

Therefore X and Y are are positively correlated.

For independent variables correlation is equal 0,but $corr(X,Y)>0$,

therefore X and Y are dependent.

Answer: (d). X and Y are positively correlated and dependent.