Download Lecture 18

AMS 311
March 30, 2000
Homework due April 6:
Chapter Six: P225: 4, 6*, 8; p232: 1, 3; p241: 2, 4, 8, 10, 16*.
On to continuous distributions
Probability density function (pdf): does not give probabilities; integrate pdf to get
probability
Cumulative distribution function (cdf)
Relation between cdf and pdf
Definition of Expected Value
If X is a continuous random variable with probability density function f, the expected

value of X is defined by E ( X ) 
 xf ( x)dx, provided that the integral converges

absolutely.
Example
c
,    x   , is called a
1 x2
Cauchy random variable. Find c so that the f(x) is a pdf. Show that E(X) does not exist.
A random variable X with density function f ( x ) 
Don’t be bashful about checking your old calculus books and tables of integrals! From
there, you will find
dx
 1  x 2  arctan x.
Theorem 6.2 is used to prove Theorem 6.3 (The law of the unconscious statistician).
Theorem 6.2.
For any continuous random variable X with probability distribution function F and
density function f,


0
0
E ( X )   [1  F (t )]dt   F ( t )dt .
Law of the unconscious statistician.
Theorem 6.3.
Let X be a continuous random variable with probability density function f(x); then for any
function h: RR,

E (h( X )) 
 h( x) f ( x)dx.

Example:
1
, a  x  b, and zero otherwise be the pdf of the random variable X.
b a
Find E (e tX ).
Let f ( x ) 
This theorem also is the basis for proving that expectation is a linear operator for sums of
functions of X.
Definition of var (X)
The variance of the random variable X is still var( X )  E ( X  EX ) 2 .
7.1. Uniform Distribution
The distribution in the example above is called a uniform random variable over (a, b).
Show that f(x) is a pdf.
Find the cdf.
Find E(X).
Find var(X).