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\centerline{Mathematics 375 -- Probability and Statistics 1}
\centerline{Discussion 2 -- The ``Method of Moment-Generating Functions"}
\centerline{December 2, 2011}
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{\it Background}
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Because of the Uniqueness Theorem (Theorem 6.1 in our text), whenever we can compute
the moment-generating function for a random variable and recognize
it as one of our standard forms, then we know its distribution.  As a result, 
we know its probability density function, mean, variance, and
hence ``everything about it''(!)  Today, we want to use this 
idea to work several examples and identify what we have. 
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\noindent
Here are two useful facts:
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\item{1.}  From a problem set, we know that if $U = a Y + b$, 
where $a,b$ are constants, then 
$$m_U(t) = e^{bt} m_Y(at)$$
\item{2.}  If $Y_1,\ldots,Y_n$ are independent and $U = Y_1 + \cdots + Y_n$,
then
$$m_U(t) = \prod_{i=1}^n m_{Y_i}(t) = m_{Y_1}(t) \times \cdots \times m_{Y_n}(t)$$ 
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{\it Discussion Questions}
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A) Prove ``useful fact 2'' above in the case that $Y_1,\ldots,Y_n$ are
jointly continuous by writing out $E(e^{tU})$ in terms
of $Y_1,\ldots,Y_n$.  Use the fact that if $Y_1,\ldots,Y_n$ are independent, 
then their joint density has the form
$$f(y_1,\ldots,y_n) = f_1(y_1) \times \cdots \times f_n(y_n),$$
where $f_i(y_i)$ are the marginal densities.
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\noindent
B)  Next, we will verify one point that
we deferred in discussing the use of the standard normal table.
Recall, we said that if $Y$ is normal with mean $\mu$ and 
standard deviation $\sigma$, then 
$$Z = {Y - \mu\over\sigma}$$
would have a standard normal distribution (i.e. normal with
mean 0 and standard deviation 1).  We never actually justified this 
claim before, except in a review problem for Exam 2.  
Note that 
$$Z = {1\over\sigma} Y - {\mu\over\sigma}.$$
Find the moment generating function of $Z$ given the
moment generating function for the normal $Y$:
$$m_Y(t) = e^{{t^2\sigma^2\over 2} + \mu t}.$$
and the ``useful fact'' 1 above.  Deduce
that $Z$ must have a standard normal distribution.
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C)  In class we showed that if $Z \sim Normal(0,1)$ is a standard
normal, then $Z^2$ has a $\chi^2$ distribution with one degree of 
freedom, by computing its moment-generating function.  
Suppose $Z_1, \ldots, Z_k$ are independent random
variables, each with a standard normal distribution.
\item{1.}  What
is the distribution of $X = Z_1 + \cdots + Z_k$?  
\item{2.}  What
is the distribution of $U = Z_1^2 + \cdots + Z_k^2$?
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{\it Assignment}
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Group writeups due in class by Monday, December 5.
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