<s>[INST] Expectation of joint probability mass function Let the joint probabilty mass function of discrete random variables X and Y be given by
$f(x,y)=\frac{x^2+y^2}{25}$, for $(x,y) = (1,1), (1,3), (2,3)$
The value of E(Y) is  ?
Attempt
$E(Y) = \sum_{x,y} y\cdot\frac{x^2 + y^2}{25}$
$E(Y) = \sum_{x,y}\frac{x^2y + y^3}{25}$
Substituting for $(x,y) = (1,1), (1,3), (2,3)$
$E(Y) = \frac1{25} + \frac{30}{25} + \frac{39}{25}$
$E(Y) = 2.80$
Is this right?
 [/INST] \begin{align}
\mathbb E[Y] &= \sum_y y\cdot\mathbb P(Y=y)\\
&= 1\cdot\mathbb P(Y=1) + 3\cdot\mathbb P(Y=3)\\
&= \frac{1^2+1^2}{25} + 3\left(\frac{1^2+3^2}{25}+\frac{2^2+3^2}{25} \right)\\
&= \frac{71}{25}.
\end{align}
</s>