1. 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 f X (x) and is called the density function.  This function f X (x) is used to calculate an area that represents the probability that X assumes a value in [x 1,x 2 ].

2. 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, f X (x) satisfies: 1) f X (x)  0 2) -    f X (x)dx = 1 3) P(x 1  X  x 2 ) = x1  x2 f X (u)du  If X is a continuous RV, then for any x 1 and x 2, P(x 1  X  x 2 ) = P(x 1 <X  x 2 ) = P(x 1  X<x 2 ) = P(x 1 <X<x 2 )

3. Probability & Statistics I Cumulative Distribution Functions  The cumulative distribution function of a continuous RV “X”, denoted by F x (x), is  F X (x) = P(X  x) = -   x f X (u)du for -  <x< 

4. 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 f X (x)  <x<  The mean of X =  x = E(X) = -    x f X (x)dx  The variance of RV “X” is denoted as  2 X or V(X).   2 X = V(X) = E(X -  x ) 2 = - -    (x -  x ) 2 f X (x)dx   X =   2 X (standard deviation = + square root of variance)

5. Probability & Statistics I Summary of Continuous Distributions  Continuous Uniform Distribution  Normal Distribution  Normal Approximation to Binomial and Poisson Distributions  “Six Sigma” Quality  Exponential Distribution

6. 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!