chapter 6 continuous probability...

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

.10

.20

.30

.40

.50

0 1 2 3 4 Values of Random Variable x (TV sales)

Pro

bab

ilit

y

Continuous Probability Distributions

Uniform Probability Distribution

Normal Probability Distribution

Exponential Probability Distribution

f (x)

x

Uniform

x

f (x) Normal

x

f (x) Exponential

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)

x

Uniform

x1 x2

x

f (x) Normal

x1 x2

x1 x2

Exponential

x

f (x)

x1 x2

Uniform Probability Distribution

f (x) = 1/(b – a) for a < x < b = 0 elsewhere

A random variable is uniformly distributed whenever

the probability is proportional to the interval’s length.

The uniform probability density function is:

f (x)

x

•where: 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

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 Density Function

f(x) = 1/10 for 5 < x < 15

= 0 elsewhere

where:

x = salad plate filling weight

Uniform Probability Distribution

Uniform Probability Distribution

Uniform Probability Distribution

for Salad Plate Filling Weight

f(x)

x 5 10 15

1/10

Salad Weight (oz.)

f(x)

x 5 10 15

1/10

Salad Weight (oz.)

What is the probability that a customer

will take between 12 and 15 ounces of salad?

12

Uniform Probability Distribution

Notice, we simply used the formula for the area of a rectangle, BASE *

HEIGHT

Var(x) = (b - a)2/12

E(x) = (a + b)/2

Uniform Probability Distribution Expected Value of x

Variance of x

Expected Value of x

Variance of x

E(x) = (a + b)/2

= (5 + 15)/2

= 10

Var(x) = (b - a)2/12

= (15 – 5)2/12

= 8.33

Uniform Probability Distribution

Normal Probability

Distribution 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

under a curve. We simply

are ____________

________ of the curve

Normal curve will be

used extensively

throughout the rest of

this semester and next

semester.

Heights of people

Scientific measurements

Test scores

Amounts

of rainfall

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

_______________

-10 0 20

x

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”

Interpretation of z

◦ The number of ________________x is

from the mean

zx

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

P(45<X<60) = P( < < ) X

CONVERT TO A STATEMENT ABOUT Z

Draw in your z values

Go to table

Find area of interest.

______________________________

Answer = ____________

z = 1 z = -.5

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

Think Plotting Points

where: = mean

e = 2.71828

/1)( xexf For x ≥ 0, > 0

0.0000

0.0500

0.1000

0.1500

0.2000

0.2500

0.3000

0.0 4.0 8.0 12.0 16.0 20.0 24.0 28.0

4

1/

Total area under curve is 1.0

where: = mean

e = 2.71828

/1)( xexf

Exponential Distribution

• Variable is quantitative ( continuous).

• X values must be positive (or zero).

• Only one parameter: (mean)

• Std. deviation: = (same value)

SKEWED right.

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

◦ This formula gives the area to the ______ of

x0

◦ X0 is some specific value of the variable x

/01)(x

o exxP

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.

Remember in Excel:

◦ Lambda = 1/exponential

Final Point on Exponential