Question :

Let $\displaystyle X_1$ and $\displaystyle X_2$ be the joint probability distribution

$\displaystyle . \qquad \qquad \qquad \qquad X_1=0$ $\displaystyle . \qquad \qquad \qquad X_1=1$ $\displaystyle . \qquad \qquad \qquad X_1=2$

$\displaystyle X_2 = 0$ $\displaystyle . \qquad \qquad \qquad 0.1$ $\displaystyle . \qquad \qquad \qquad \qquad 0.4$ $\displaystyle . \qquad \qquad \qquad \qquad 0.1$

$\displaystyle X_2 = 1 $ $\displaystyle . \qquad \qquad \qquad 0.2$ $\displaystyle . \qquad \qquad \qquad \qquad 0.2$ $\displaystyle . \qquad \qquad \qquad \qquad 0.0$

i) Find P(X_1 + X_2)

ii) Are X_1 and X_2 independent

iii) P(X_1 = 0 | X_2 = 1)

I need to know how to get the value of $\displaystyle P(X_1 + X_2)$

Here is my work

$\displaystyle

\begin{array}{|c|c|c|} P(X_1 = 0) \ = \ 0.3 \ \ & P(X_1 = 0 |X_2 = 0) \ = \ 0.2 \ \ & P(X_1 = 0 | X_2 = 1) \ = \ 0.5 \\

P(X_1 = 1) \ = \ 0.6 \ \ & P(X_1 = 1 |X_2 = 0) \ = \ 0.7 \ \ & P(X_1 = 1 | X_2 = 1) \ = \ 0.5 \\

P(X_1 = 2) \ = \ 0.1 \ \ & P(X_1 = 2 |X_2 = 0) \ = \ 0.2 \ \ & P(X_1 = 2 | X_2 = 1) \ = \ 0.0 \\

\end{array}

$

And also want to know the if my answers of ii) and iii) are correct or no ?

i) ???

ii) Answer

Since $\displaystyle P(X_1|X_2) = P(X_1) \ \therefore \ P(X_1) \ and \ P(X_2)$ are not independent

iii) Answer

$\displaystyle

P(X_1 = 0| X_2 = 1) \ = \ 0.5$

Please provide me with the answer for i) ?????????