Random trial $E$ has three pairs of incompatible results $A_{1}$,$A_{2}$,and $A_{3}$, and the probability of the three results is $\frac{1}{3}$. If test $E$ is repeated twice independently,$X$ represents the number of times of the result $A_{1}$ in the two trials, and $Y$ represents the number of times of the result $A_{2}$ in the two trials, then the correlation coefficient of $X$ and $Y$ is？
My problem is a basic question.
why $EX=EY=\frac{2}{3}$ obvious?
why $DX=DY=2\times \frac{1}{3}\times\frac{2}{3}=\frac{4}{9}$ obvious?
I am very grateful for any tips and help.
$X$ is a binomial random variable with parameters $p = 1/3$ and $N = 2$. We know that the expected value of a binomial random variable is $$ E(X) = Np = 2 \cdot \frac13. $$
There is also a standard formula for the variance of a binomial random variable.