Random variabe $ X $ has discrete uniform distribution $ P(X=i) = {1 \over{100}} , i \in \{1,2,...100\}$.
Let random variables $ Y $ and $ Z $ be defined by
$ Y = 1 $ when $2|X$ or $3|X$. Otherwise $Y = 0$.
$ Z = 1 $ when $3|X$. Otherwise $Z = 0$.
Find correlation coefficient $p$ of variables $Y$ and $Z$.
I know that $p = {E [(Y - E(Y))(Z - E(Z)) \over { \sqrt{V(Y)V(Z)}}} $, but I have know idea how to find out density or distribution of $Y$ and $Z$ needed to calculate the correlation coefficient of this variables.
$Y$ and $Z$ only take values in $\{0,1\}$ so that:$$\mathbb EY=0\cdot P(Y=0)+1\cdot P(Y=1)=P(Y=1)$$and likewise $\mathbb EZ=P(Z=1)$.
Further $Y=Y^2$ and $Z=Z^2$ so above we also found $\mathbb EY^2$ and $\mathbb EZ^2$.
Then variances can be found on base of the rule $\mathsf{Var}(X)=\mathbb EX^2-(\mathbb EX)^2$.
Now it remains to find the covariance.
We can apply $\mathsf{Cov}(Y,Z)=\mathbb EYZ-\mathbb EY\mathbb EZ$ and already have $\mathbb EY\mathbb EZ$ so it remains to find $\mathbb EYZ$.
Now again note that $YZ$ only take values in $\{0,1\}$ so that $\mathbb EYZ=P(YZ=1)=P(Y=1\wedge Z=1)$.