AMS 311
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 [pic] provided that the integral converges absolutely.
Example
A random variable X with density function [pic] is called a Cauchy random variable. Find c so that the f(x) is a pdf. Show that E(X) does not exist.
Don’t be bashful about checking your old calculus books and tables of integrals! From there, you will find
[pic]
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,
[pic]
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,
[pic]
Example:
Let [pic] and zero otherwise be the pdf of the random variable X. Find [pic]
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 [pic]
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).
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