Sums of discrete random variables
Problem 03 page 289
Let \(X_{1}\) and \(X_{2}\) be independent random variables with the same distribution
\[p_{X} = \left\{\begin{matrix} \frac{1}{8} \text{ ; if }x=0\\ \frac{3}{8} \text{ ; if }x=1\\ \frac{1}{2} \text{ ; if }x=2 \end{matrix}\right.\] Find the distribution of \(S_{2} = X_{1} + X_{2}\)
Solution:
\[P(S_{2}=0)=P(x_{1}=0)+P(x_{2}=0)=\frac{1}{8}*\frac{1}{8}=\frac{1}{64} \]
\[P(S_{2}=1)=P(x_{1}=0)+P(x_{2}=1)+P(x_{1}=1+P(x_{2}=0)=2 * \frac{1}{8}*\frac{3}{8}=\frac{3}{32} \] \[P(S_{2}=2)=P(x_{1}=0)+P(x_{2}=2)+P(x_{1}=2)+P(x_{2}=0)+P(x_{1}=1)+P(x_{2}=1)=2 * \frac{1}{8}*\frac{1}{2}+\frac{3}{8}*\frac{3}{8}=\frac{17}{64} \]
\[P(S_{3}=2)=P(x_{1}=1)+P(x_{2}=2)+P(x_{1}=2)+P(x_{2}=1)=2 * \frac{3}{8}*\frac{1}{2}=\frac{3}{8} \]
\[P(S_{2}=4)=P(x_{1}=2)+P(x_{2}=2)=\frac{1}{2}*\frac{1}{2}=\frac{1}{4} \]