Some Useful Continuous Probability Distributions

8.1 Properties of Continuous Probability DistributionsA smooth curve known as the density function, is used to represent the probability distribution of a continuous random variable

The curve must never fall below the x-axis … for all x

The total area under the curve must be 1 …

xf

0xf

1 xf

For continuous random variables we assign probability to intervals. (Not Points)

the area under the curve between a and b

With continuous variables, each point has probability zero

Thus for continuous variables

)( bxaP

0)( axP0)( bxP

)()( bxaPbxaP

For continuous distributions

Population Mean =

Population Variance =

Population Standard Deviation =

xxf

2222 xfxxfx

2

8.2 The Uniform DistributionThe density function for the uniform distribution is

as follows: for

Calculating descriptive statistics

Population Mean =

Population variance =

Population standard deviation =

ab

xf

1 bxa

2

ba

12

22 ab

2

The probability that a value is between c and d is

where

Example: Travel time from Lexington KY to Columbus OH is a uniform distributed between 200 and 240 minutes

Give the density functionFind the mean.Find the median.Find the variance.Find the standard deviation.Find the probability of arriving in less than 225

minutes.

ab

cddxcP

bdca

8.3 The Normal Distribution

Density function for the normal distribution is as follows:

2

2

2

22

1

x

exf

The normal distribution is a common type of continuous distribution. It is a bell shaped curve.

The bell is symmetric about the mean of the random variable .

The standard deviation of the random variable measures the spread of the bell. The larger is the more spread out the bell.

For the normal the mean, median, and mode are equal.

The value of and characterize which normal distribution we are using

The normal distribution with and is called the standard normal distribution. (This is used to calculate probabilities for all normal distributions)

If X is normal with mean and standard deviation ,

then is standard normal.

0 1

x

Z

ExamplesDraw Pictures of desired areas when doing

problems!!!

Facts:Total area under the curve is 1Curve is symmetric about 0

Combining these facts with the table allows us to compute all probability statements for Z

21)0()0( ZPZP

)55.10( ZP

)96.10( ZP

Examples

Notice that Probabilities in the table stop at 3.9 with an area of .5000. Beyond this Z value you will always have close to .5 the area.

64.1ZP

)64.1(ZP

)64.1(ZP

)032.2( ZP

)22( ZP

)18.241.1( ZP

8.4 Calculating Areas Under Any Normal CurveIf x is normal with mean and standard

deviation ,

then is standard normal.

Write probability statement for XRewrite in terms of Z

x

Z

ExampleThe distribution of IQ scores for the general population is approximately normal with and .

x = IQ score of randomly selected person

Find

Find

10010

)120100( XP

)130( XP

ExampleSuppose the amount of Pepsi in a “12 oz” can has a Normal distribution with and X = amount of Pepsi in a Randomly selected can

Find

oz. 12 oz. 1.

)83.11( XP