Discrete Probability Distributions

Example: Let $ X$ be the random variable: number of heads in 3 tosses of a fair coin.

Sample Space $ x$
TTT 0
TTH 1
THT 1
THH 2
HTT 1
HTH 2
HHT 2
HHH 3
 
$ P(X=x)$: Probability that outcome is a specific $ x$ value.
x 0 1 2 3
P(X=x) $ \frac{1}{8}$ $ \frac{3}{8}$ $ \frac{3}{8}$ $ \frac{1}{8}$

\begin{displaymath}
f(0)=P(X=0)=\frac{
\left(
\begin{array}{c}
3\\
0\\
\end...
...\begin{array}{c}
8\\
2\\
\end{array}\right)}=\frac{10}{28}
\end{displaymath}

\begin{displaymath}
f(1)=P(X=1)=\frac{
\left(
\begin{array}{c}
3\\
1\\
\end...
...\begin{array}{c}
8\\
2\\
\end{array}\right)}=\frac{15}{28}
\end{displaymath}

\begin{displaymath}
f(2)=P(X=2)=\frac{
\left(
\begin{array}{c}
3\\
2\\
\end...
...
\begin{array}{c}
8\\
2\\
\end{array}\right)}=\frac{3}{28}
\end{displaymath}

\begin{displaymath}
\begin{array}{l}
f(0)=\frac{1}{16},f(1)=\frac{4}{16},f(2)=\...
...frac{15}{16}\\
F(4)=f(0)+f(1)+f(2)+f(3)+f(4)=1\\
\end{array}\end{displaymath}

\begin{displaymath}
F(x)=\left\lbrace
\begin{array}{c}
0 ~ for~ x< 0\\
\frac...
...q x < 4 \\ \\
1~ for~ x \geq 4 \\
\end{array}\right\rbrace
\end{displaymath}

Figure 1: Bar chart and probability histogram
\includegraphics[scale=0.45,angle=1]{figures/03-01} \includegraphics[scale=0.45]{figures/03-02}

Figure 2: Discrete cumulative distribution.
[
\includegraphics[scale=0.45]{figures/03-03}

Cem Ozdogan 2010-03-15