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\centerline{Mathematics 375 -- Probability and Statistics 1}
\centerline{Problem Set 7 -- {\it due}: Wednesday, November 5, 2003}
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From the text:  Chapter 4/69, 70, 71, 72, 75, 83, 87, 91, 97, 98, 101.
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{\it Additional Problems}
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In most cases, the {\it only} way to compute probabilities 
with Gamma- and Beta-distributed random variables is to use
numerical methods on the appropriate integrals of the pdf's.
However, in some special cases, there are interesting connections
with some of the discrete random variables we have studied
before. 
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A)  In this problem you will study a relation between continuous
Gamma distributions and discrete Poisson distributions in the case
that the parameter $\alpha$ of the Gamma distribution is a positive
integer and $\beta = 1$.
\item{1)}  Let $\alpha$ be a positive integer and 
consider a Gamma-distributed random variable $Y$ with parameters
$\alpha$ positive $\in {\bf Z}$, and $\beta = 1$.
Let $\lambda > 0$ be arbitrary.  Using integration by parts
and a proof by mathematical induction, show that 
$$\eqalign{P(Y \ge \lambda) &= 
{1\over \Gamma(\alpha)} \int_\lambda^\infty y^{\alpha - 1} e^{-y}\ dy\cr
&= \sum_{n=0}^{\alpha-1} {\lambda^n\over n!} e^{-\lambda}\cr}$$
(Note that the last line here is a sum of probabilities
for a Poisson discrete random variable(!)).  
\item{2)}  Suppose $Y$ has a Gamma distribution with 
$\alpha = 3$, $\beta = 1$.  Using the result of part 1 and
the tables in our text, find $P(Y \ge 5)$.
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B)  In this problem you will study a relation between continuous
Beta distributions and discrete binomial distributions in the case
that the parameters $\alpha,\beta$ of the Beta distribution are positive
integers.
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\item{1)}  Let $Y$ be a continuous random variable with a Beta
distribution where $\alpha$ and $\beta$ are positive integers.
Let $0 < y < 1$.  Show that 
$$\eqalign{P(0 \le Y \le p) &= {1\over B(\alpha,\beta)}
\int_0^p y^{\alpha-1} (1-y)^{\beta -1}\ dy\cr
&= \sum_{n = \alpha}^{\alpha+\beta-1} 
{\alpha+\beta-1\choose n}p^n(1-p)^{\alpha+\beta-1-n}\cr}$$
(Note that this is a sum of binomial probabilities(!)).
\item{2)}  Suppose that $Y$ has a Beta distribution with 
$\alpha = 4$ and $\beta = 7$.  Use the result of part 1 and 
the tables in the text to find $P(0 \le Y \le .3)$.
\bye
