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\date{February 8, 2005}

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\title{Stat 324: Lecture 07\\
Binomial distributions}

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\author{Moo K. Chung\\
mchung@stat.wisc.edu}

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\begin{enumerate}



\item Consider an experiment that give only two outcomes $H$ or $T$.
Let $P(H)=p$ and $P(T)=1-p$. This is a Bernoulli experiment. Now
consider $n$ independent experiments where each experiment gives
only two outcomes. This is a {\em Binomial experiment}. Let $X$ be
the number of outcome $H$. This is called the {\em Binomial random
variable} with parameter $n$ and $p$. It will be denoted by $X \sim
Binomial(n,p)$. Note that if $X_i$ is the number of outcome $H$ in
the $i$-th experiment, $X=X_1 + X_2 + \cdots + X_n$.

\item {\em Binomial distribution}. The distribution is given by
$P(X=x) = {n \choose x}p^x(1-p)^{n-x}$. $$\mathbb{E}X=np,
\mathbb{V}X=np(1-p).$$
\begin{verbatim}
> plot(dbinom(0:10,10,0.5),type="s")
> plot(pbinom(0:10,10,0.5),type="s")
\end{verbatim}

\begin{figure}[h]
\centering
\includegraphics[width=50mm]{324.07.02.ps}
\includegraphics[width=50mm]{324.07.01.ps}
\caption{Left: distribution of $Binomial(10,0.5)$, Right: cdf of
$Binomial(10,0.5)$}\label{fig:hist}
\end{figure}
\item A random variable is {\em continuous} if its set of possible
values is an entire interval of numbers. For a continuous random
variable $X$, a probability distribution (density) of $X$ is a
function $f$ that satisfies
$$P(a \leq X \leq b) = \int_a^b f(x)\;dx$$
for  any $a$ and $b$. The {\em cumulative distribution function} $F$
of a continuous random variable $X$ is defined as
$$F(x) \leq P(X \leq x) = \int_{-\infty}^x f(x) \;dx.$$
From the fundamental theorem of calculus, the density function $f$
can be obtained as $f(x)=F'(x)$.
\item For given $p$, the $100p$-th {\em percentile} of the distribution of $X$ is a number
$\eta(p)$ such that
$$p = \int_{-\infty}^{\eta(p)} f(x) \;dx.$$
\end{enumerate}
 {\bf Assigned Problems} Exercise 3.56. Homework II due on
Feb 10. 9:30AM. Submit solutions to assigned problems from Lecture
4-7 (7 problems). Do the following problem for Homework III.
Exercise 4.6.


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