Chapter3.lyx.emergency

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\begin_layout Subsection*
3.8 Functions of a Random Variables
\end_layout

\begin_layout Standard

\series bold
Theorem 3.8.1 
\series default
Function of a Discrete Random Variable.
 Let 
\begin_inset Formula $X$
\end_inset

 have a discrete distribution with p.d.
 
\begin_inset Formula $f$
\end_inset

, and let 
\begin_inset Formula $Y=r(X)$
\end_inset

 for some function of 
\begin_inset Formula $r$
\end_inset

 defined on the set of possible values of 
\begin_inset Formula $X$
\end_inset

.
 For each possible value 
\begin_inset Formula $y$
\end_inset

 of 
\begin_inset Formula $Y$
\end_inset

, the p.f.f 
\begin_inset Formula $g$
\end_inset

 of 
\begin_inset Formula $Y$
\end_inset

 is :
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
g(y)=Pr(Y=y)=Pr[r(X)=y]=\sum_{x:r(x)=y}f(x)
\]

\end_inset


\end_layout

\begin_layout Subsection*
3.9 Functions of Two or More Random Variables.
\end_layout

\begin_layout Standard
The general method for solving problems like those of Example 3.9.1 is a straight
 forwatd extension of 
\series bold
Theorem 3.8.1.
 
\end_layout

\begin_layout Standard

\series bold
Theorem 3.9.1 
\series default
Functions of Discrete Random Variables.
 Suppose that 
\begin_inset Formula $n$
\end_inset

 random variables 
\begin_inset Formula $X_{1},...,X_{n}$
\end_inset

 have a discrete joint distribution for which the joint p.f.
 is 
\begin_inset Formula $f$
\end_inset

, and that 
\begin_inset Formula $m$
\end_inset

 functions 
\begin_inset Formula $Y_{1},...,Y_{m}$
\end_inset

 of these 
\begin_inset Formula $n$
\end_inset

 random variables are defined as follows:
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
Y_{1}=r_{1}(X_{1},...,X_{n}),...,Y_{m}=r_{m}(X_{1},...,X_{n})
\]

\end_inset


\end_layout

\begin_layout Standard
For given values 
\begin_inset Formula $y_{1},...,y_{m}$
\end_inset

 of the 
\begin_inset Formula $m$
\end_inset

 random variables 
\begin_inset Formula $Y_{1},...,Y_{m}$
\end_inset

, let 
\begin_inset Formula $A$
\end_inset

 denote the set of all points 
\begin_inset Formula $(x_{1},...,x_{n})$
\end_inset

 such that 
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
r_{1}(x_{1},...,x_{n})=y_{1};...;r_{m}(x_{1},...,x_{n})=y_{m}.
\]

\end_inset


\end_layout

\begin_layout Standard
Then the value of the joint p.f.
 
\begin_inset Formula $g$
\end_inset

 of 
\begin_inset Formula $Y_{1},...,Y_{m}$
\end_inset

 is specified at the point 
\begin_inset Formula $(y_{1},...,y_{m})$
\end_inset

 by the relation
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
g(y_{1},...,y_{m})=\sum_{(x_{1},...,x_{n})\in A}f(x_{1},...,x_{n})
\]

\end_inset


\end_layout

\begin_layout Standard

\series bold
Theorem 3.9.2 
\series default
Binomial and Bernoulli Distributions Assume that 
\begin_inset Formula $X_{1},...,X_{n}$
\end_inset

 are i.i.d.
 random variables having the Bernoulli distribution with parameter 
\begin_inset Formula $p$
\end_inset

.
 Let 
\begin_inset Formula $Y=X_{1}+...+X_{n}$
\end_inset

.
 Then 
\begin_inset Formula $Y$
\end_inset

 has the binomial distribution with parameters 
\begin_inset Formula $n$
\end_inset

 and 
\begin_inset Formula $p$
\end_inset

.
\end_layout

\begin_layout Standard

\series bold
Question: 
\series default
In Binomial case is not equally likely intuition, so explain how it different
 with equally likely as 
\series bold
Example 3.9.2.
\end_layout

\begin_layout Subsection*
Random Variables with a Continuous Joint Distribution 
\end_layout

\begin_layout Standard
Example 3.9.4 is an example of a brute-force method that is always availabel
 for finding the distribution of a function of several random variables,
 however, it might be difficult to apply in individual cases.
\end_layout

\begin_layout Paragraph*
Theorem 3.9.3 
\series medium
Brute-Force Distribution of a Function.
 Suppose that the joint p.d.f of 
\begin_inset Formula $X=(X_{1},...,X_{n})$
\end_inset

 is 
\begin_inset Formula $f(\boldsymbol{x})$
\end_inset

 and that 
\begin_inset Formula $Y=r(\boldsymbol{X}).$
\end_inset

 For each real number 
\begin_inset Formula $y$
\end_inset

, define 
\begin_inset Formula $A_{y}=\{\boldsymbol{x:}r(\boldsymbol{x})\leq y\}.$
\end_inset

 Then the c.d.f 
\begin_inset Formula $G(y)$
\end_inset

 of 
\begin_inset Formula $Y$
\end_inset

 is 
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
G(y)=\int\ldots_{A_{y}}\int f(x)dx
\]

\end_inset


\end_layout

\begin_layout Standard
If the distribution of 
\begin_inset Formula $Y$
\end_inset

 also is continuous, then the p.d.f of 
\begin_inset Formula $Y$
\end_inset

 can be found by differentiating the c.d.f.
 
\begin_inset Formula $G(y)$
\end_inset

.
\end_layout

\begin_layout Standard
A popular special case of 
\series bold
Theorem 3.9.3 
\series default
is the following.
\end_layout

\begin_layout Paragraph*
Theorem 3.9.4 Linear Function of Two Random Variables.
 
\series medium
Let 
\begin_inset Formula $X_{1}$
\end_inset

 and 
\begin_inset Formula $X_{2}$
\end_inset

 have joint p.d.f 
\begin_inset Formula $f(x_{1},x_{2})$
\end_inset

, and let 
\begin_inset Formula $Y=\alpha_{1}X_{1}+\alpha_{2}X_{2}+b$
\end_inset

 with 
\begin_inset Formula $a_{1}\neq0$
\end_inset

.
 The n 
\begin_inset Formula $Y$
\end_inset

 has a continous distribution whose p.d.f is 
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
g(y)=\int_{-\infty}^{\infty}f(\frac{y-b-a_{2}x_{2}}{a_{1}},x_{2})\frac{1}{|a_{1}|}dx_{2}
\]

\end_inset


\end_layout

\begin_layout Paragraph*
Definition 3.9.1 Convolution.
 
\series medium
Let 
\begin_inset Formula $X_{1}$
\end_inset

 and 
\begin_inset Formula $X_{2}$
\end_inset

 be independent continuos random variables and let 
\begin_inset Formula $Y=X_{1}+X_{2}$
\end_inset

.
 The distribution of 
\begin_inset Formula $Y$
\end_inset

 is called the convolution of the distributions of 
\begin_inset Formula $X_{1}$
\end_inset

and 
\begin_inset Formula $X_{2}$
\end_inset

.
 The p.d.f of 
\begin_inset Formula $Y$
\end_inset

 is sometime called the convolution of the p.d.f.'s of 
\begin_inset Formula $X_{1}$
\end_inset

 and 
\begin_inset Formula $X_{2}$
\end_inset

.
\end_layout

\begin_layout Standard
If we let the p.d.f of 
\begin_inset Formula $X_{i}$
\end_inset

 be 
\begin_inset Formula $f_{i}$
\end_inset

 for 
\begin_inset Formula $i=1,2$
\end_inset

 in definition 3.9.1, then Theorem 3.9.4 (with 
\begin_inset Formula $a_{1}=a_{2}=1$
\end_inset

 and 
\begin_inset Formula $b=0$
\end_inset

) says that the p.d.f of 
\begin_inset Formula $Y=X_{1}+X_{2}$
\end_inset

 is
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
g(y)=\int_{-\infty}^{\infty}f_{1}(y-z)f_{2}(z)dz
\]

\end_inset


\end_layout

\begin_layout Paragraph*

\series medium
A popular way to describe how spread out is a random sample is to use the
 distance from the minimum to the maximum, which is called the range of
 the random sample.
 We can combine the result from the end of 
\series default
Example 3.9.6 
\series medium
with 
\series default
Theorem 3.9.4 
\series medium
to find the p.d.f of the range.
\end_layout

\begin_layout Standard
\begin_inset Graphics
	filename pasted1.png

\end_inset


\end_layout

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