Download CHAPTER 6 CONTINUOUS PROBABILITY DISTRIBUTIONS

Continuous Probability Distributions

The Uniform Distribution
a



b
The Normal Distribution
The Exponential Distribution


Slide 1
The Uniform Probability Distributions
a

x1 x2
P(x1 ≤ x≤ x2)
b
a
x1

b
P(x≤ x1)
P(x≥ x1)= 1- P(x<x1)
a
x1 
P(x≥ x1)
b
Slide 2
The Uniform Probability Distribution
Uniform Probability Density Function
f (x) = 1/(b - a) for a < x < b
= 0 elsewhere
where
a = smallest value the variable can assume
b = largest value the variable can assume

The probability of the continuous random variable
assuming a specific value is 0.
P(x=x1) = 0
Slide 3
The Normal Probability Density Function
1
 ( x   )2 / 2 2
f ( x) 
e
2 
where
 = mean
 = standard deviation
 = 3.14159
e = 2.71828
Slide 4
The Normal Probability Distribution

Graph of the Normal Probability Density Function
f (x )

x
Slide 5
The Standard Normal Probability Density Function
where
 =0
 =1
 = 3.14159
e = 2.71828
Slide 6
Given any positive value for z, the table will
give us the following probability
The table will give this
probability
Given positive z
The probability that we find using the table is
the probability of having a standard normal
variable between 0 and the given positive z.
Slide 7
Given z = .83 find the probability
z
.00
.01
.02
.03
.04
.05
.06
.07
.08
.09
.0 .0000 .0040 .0080 .0120 .0160 .0199 .0239 .0279 .0319 .0359
.1 .0398 .0438 .0478 .0517 .0557 .0596 .0636 .0675 .0714 .0753
.2 .0793 .0832 .0871 .0910 .0948 .0987 .1026 .1064 .1103 .1141
.3 .1179 .1217 .1255 .1293 .1331 .1368 .1406 .1443 .1480 .1517
.4 .1554 .1591 .1628 .1664 .1700 .1736 .1772 .1808 .1844 .1879
.5 .1915 .1950 .1985 .2019 .2054 .2088 .2123 .2157 .2190 .2224
.6 .2257 .2291 .2324 .2357 .2389 .2422 .2454 .2486 .2518 .2549
.7 .2580 .2612 .2642 .2673 .2704 .2734 .2764 .2794 .2823 .2852
.8 .2881 .2910 .2939 .2967 .2995 .3023 .3051 .3078 .3106 .3133
.9 .3159 .3186 .3212 .3238 .3264 .3289 .3315 .3340 .3365 .3389
Slide 8
The Exponential Probability Distribution

Exponential Probability Density Function
f ( x) 
where

1

e x /
for x > 0,  > 0
 = mean
e = 2.71828
Cumulative Exponential Distribution Function
P( x  x0 )  1  e  x0 / 
where
x0 = some specific value of x
Slide 9
Example
The time between arrivals of cars at Al’s Carwash
follows an exponential probability distribution with a
mean time between arrivals of 3 minutes. Al would like
to know the probability that the time between two
successive arrivals will be 2 minutes or less.
P(x < 2) = 1 - 2.71828-2/3 = 1 - .5134 = .4866
Slide 10
Example: Al’s Carwash

Graph of the Probability Density Function
F (x )
.4
.3
P(x < 2) = area = .4866
.2
.1
x
1
2
3
4
5 6
7
8
9 10
Slide 11