probability & statistics i ie 254 summer 1999 chapter 4 continuous random variables what is...

Probability & Statistics I

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

3) P(x1 X x2) = x1x2 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) = -x fX(u)du for -<x<

Probability & Statistics I

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).

2X = V(X) = E(X - x)2 = - -

(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!