Download Chapter 6 Continuous Probability Distributions

L06
Chapter 6: Continuous Probability Distributions
Chapter 6 Continuous Probability
Distributions
Recall Discrete Probability Distributions
◦ Could only take on particular values
◦ Continuous can take on any value
.50
Probability

.40
.30
.20
.10
0
1
2
3
4
Values of Random Variable x (TV sales)
Continuous Probability Distributions
Uniform Probability Distribution
 Normal Probability Distribution
 Exponential Probability Distribution

f (x)
Uniform
f (x) Exponential
x f (x)
Normal
x
x
Continuous Probability Distributions




A continuous random variable can assume any value in an
interval on the real line or in intervals.
To find probabilities, we use areas under a probability
density function
It is not possible to talk about the probability of the
random variable assuming a single value.
 For example: Probability that height = 60 inches
 This is because the area under a single point is zero
Instead, we talk about the probability of the random
variable assuming a value within an _______
 For Example, Height being between 60 and 65 inches
Continuous Probability Distributions

The probability of the random variable assuming a value
within an interval from x1 to x2 is defined to be the _____
under the graph of the probability density function
between x1 and x2.
f (x)
f (x) Exponential
Uniform
f (x)
x1 x2
Normal
x1 xx12 x2
x
x1 x2
x
x
Uniform Probability Distribution

A random variable is uniformly distributed whenever
the probability is proportional to the interval’s length.

The uniform probability density function is:
f (x) = 1/(b – a) for a < x < b
=0
elsewhere
•where:
f (x)
a = smallest value the variable can assume
•
b = largest value the variable can assume
•These Statements tell us about the shape of the
probability distribution
•To find Probabilities, we need the _________
the shape
x
Uniform Probability Distribution

Example: Slater's Buffet
Slater customers are charged
for the amount of salad they take.
Sampling suggests that the
amount of salad taken is
uniformly distributed
between 5 ounces and 15 ounces.
Uniform Probability Distribution

Uniform Probability Density Function
f(x) = 1/10 for 5 < x < 15
=0
elsewhere
where:
x = salad plate filling weight
Uniform Probability Distribution

Uniform Probability Distribution
for Salad Plate Filling Weight
f(x)
1/10
5
10
15
Salad Weight (oz.)
x
Uniform Probability Distribution
What is the probability that a customer
will take between 12 and 15 ounces of salad?
f(x)
1/10
5
10 12
15
Salad Weight (oz.)
x
Notice, we simply used the formula for the area of a rectangle, BASE *
HEIGHT
Uniform Probability Distribution

Expected Value of x
E(x) = (a + b)/2

Variance of x
Var(x) = (b - a)2/12
Uniform Probability Distribution

Expected Value of x
E(x) = (a + b)/2
= (5 + 15)/2
= 10

Variance of x
Var(x) = (b - a)2/12
= (15 – 5)2/12
= 8.33
Heights
of people
Normal Probability
Distribution



Test
scores

Scientific
measurements
Is this chapter discrete or
continuous?
And how do we find the
probability of variables
that are continuous?
We are staying in the
world where we find
probability by the area
Amounts
under a curve. We simply
of rainfall
are ____________
________ of the curve
Normal curve will be
used extensively
throughout the rest of
this semester and next
semester.
Normal Probability Distribution

Let’s take a look at what the curve looks
like.
x
Normal Probability Function

Let’s take a look at the formula that
generates our curve
 = mean
 = standard deviation
 = 3.14159
e = 2.71828
Normal Probability Distribution
Characteristics
 Distribution is __________

◦ Skew is _________
◦ Tails are _____________of one another

Value on the x-axis below highest point is the
mean, median, and mode.
x
Normal Probability Distribution

Characteristics
◦ The mean can be any numerical value
◦ The mean moves the distribution to
_______________
x
-10
0
20
Normal Probability Distribution

Characteristics
◦ The standard deviation determines the
______of the curve. Greater standard
deviation, _______ the _______.
 = 15
 = 25
x
Normal Probability Distribution

Characteristics
◦ Probabilities = area under the curve.
◦ Total area =_______
◦ Area under right half = _____Same for left.
.5
.5
x
Standard Normal Distribution
There are __________ many means for a
normal distribution
 There are _______many standard deviations
for the normal distribution.
 We are going to get our probability information
from a table, but our book is not big enough to
contain infinitely man normal distribution tables.
What should we do?
 “STANDARDIZE” so we only have to use one
table

Standard Normal
Standard Normal Probability Distribution: A
normal distribution with mean of 0 and
standard deviation of 1
 All normal distributions can be
___________into the standard normal
distribution
 We use the transformation so we don’t have to
have infinitely many tables in the back of the
book.
 The letter z is used to designate the standard
normal random variable

Standard Normal

Transforming from “normal” to “standard
normal”
x
z


Interpretation of z
◦ The number of ________________x is
from the mean
Example


Now let’s work on a problem where we have to go from a normal
distribution to a standardized normal distribution.
The time required to build a computer is normally distributed with
a mean of 50 minutes and a standard deviation of 10 minutes. What
is the probability that a computer is assembled in a time between
45 and 60 minutes? Algebraically speaking, what is P(45 < X < 60)?
◦ Method:
1. Draw
2. Convert to Z
3. Look up probabilities in Table
0
Example
CONVERT TO A STATEMENT ABOUT Z
P(45<X<60) = P(
<
Draw in your z values
Go to table
Find area of interest.
______________________________
Answer = ____________
X 

<
)
z = -.5
z=1
Exponential Probability Distribution

Useful to describe _______it takes to
complete a task or for something to
happen
Time between
vehicle arrivals
at a toll booth
Time required
to complete
a questionnaire
Distance between
major defects
in a highway
Exponential Probability Distribution
We are staying in the world where we
find probability by the area under a curve.
We simply are changing the shape of
the curve
 Shape of the curve can be represented by
the density function

f ( x) 
1

e x / 
For x ≥ 0,  > 0
where:

Think Plotting Points
 = mean
e = 2.71828
f ( x) 
1

e
x / 
where:
 = mean
4
e = 2.71828
0.3000
1/

0.2500
Total area under curve is 1.0
0.2000
0.1500
0.1000
0.0500
0.0000
0.0
4.0
8.0
12.0
16.0
20.0
24.0
28.0
Exponential Distribution
•
Variable is quantitative ( continuous).
X values must be positive (or zero).
•
Only one parameter:
•
Std. deviation:
•
SK

E
=
ED right.
W

(mean)
(same value)
Exponential Probability Distribution

How to work with Exponential
◦ Uniform – we used the formula base * height
to find the area
◦ Normal – we used the table to find the area
◦ Exponential – we use a _______ to find the
area
P( x  xo )  1  e
 x0 / 
◦ This formula gives the area to the ______ of
x0
◦ X0 is some specific value of the variable x
Relationship Between Poisson and
Exponential
Poisson  Number of occurrences per interval
 Exponential  TIME between occurrences
 Skill we want. Go from Poisson to Exponential
Framework
 Number of cars that arrive at the carwash
follows a Poisson Distribution with mean of 10
cars per hour.

◦ How do we go from cars per hour to hours per car?
◦ 10 cars/hour  1 hour / 10 cars = .1 hours per car
◦ This tells us  for the exponential distribution

Skill: When given poisson, we can
______to exponential by division.
Final Point on Exponential

Remember in Excel:
◦ Lambda = 1/exponential