Transcript IE254 Summer`99

IE 254 Summer 1999 Chapter 4
 Continuous Random Variables
 What is the difference between a discrete & a
continuous R.V.?
 Probability Distributions & Density Functions
 The function which enables us to calculate probabilities
involving RV “X” is denoted as fX(x) and is called the
density function.
 This function fX(x) is used to calculate an area that
represents the probability that X assumes a value in
[x1,x2].
Probability & Statistics I
Probability Density Functions
 Think of the pdf in continuous distributions as analogous to the
pmf used in discrete distributions.
 For a random variable X, fX(x) satisfies:
1) fX(x)  0

2) - fX(x)dx = 1
x2
3) P(x1  X  x2) = x1 fX(u)du
 If X is a continuous RV, then for any x1and x2,
P(x1 X x2) = P(x1<X x2) = P(x1 X<x2) = P(x1<X<x2)
Probability & Statistics I
Cumulative Distribution Functions
 The cumulative distribution function of a continuous RV
“X”, denoted by Fx(x), is
 FX(x) = P(X  x) = -
Probability & Statistics I
x
fX(u)du
for -<x<
Expected Values of a Continuous R.V.

The mean and variance of a continuous RV are defined in a similar
fashion as a discrete RV except that integration replaces summation in
the definitions!

For continuous RV “X” with pdf fX(x) <x<

The mean of X = x= E(X) = - xfX(x)dx
The variance of RV “X” is denoted as 2X or V(X).

2
2
  X = V(X) = E(X - x) = - - (x - x)2 fX(x)dx
 X = 2X (standard deviation = + square root of variance)

Probability & Statistics I
Summary of Continuous Distributions
 Continuous Uniform Distribution
 Normal Distribution
 Normal Approximation to Binomial and Poisson
Distributions
 “Six Sigma” Quality
 Exponential Distribution
Probability & Statistics I
IE 254 Summer 1999 Chapter 4 Homework
 Homework Assignment:
Chapter 4 #’s 13, 22, 24, 25, 27, 42, 43, 45, 47, 55, 67, 68,
85, 86, 138 - 138 is for fun! (but turn it in!)
Due Friday July 9, 1999!
Probability & Statistics I