STAT3010 Lecture 1

1

Lecture 1:

REVIEW OF DISTRIBUTIONS

Continuous Distributions:

Definitions

Notes: There is no area under the density curve above a single value. (e.g. above 2.5)and P( bxa ) = P( bxa )

Why?

A density function )(xf is used to describe (at least approximately) the population or process distribution of a continuous variable x . The graph of )(xf is called the density curve. The following properties must be satisfied:

1. 0)(xf .2. 1)( dxxf (the total area under the density curve is 1.0)

3. For any two numbers a and b with ba ,

proportion of x values between a and b = ba

dxxf )( .

(This proportion is the area under the density curve and above the interval with endpoints a and b).

STAT3010 Lecture 1

2

Example: Verify the following pdf 21210,

)(xxxx

xf is valid.

NORMAL DISTRIBUTION

Definition

A continuous variable x is said to have a normal distributionwith parameters and , where and 0 , if the density function of x is

)2()( 22

21)( xexf for x

STAT3010 Lecture 1

3

Mean of the Normal Distribution?

Variance of the Normal Distribution?

THE STANDARD NORMAL DISTRIBUTION

Definition

Standard Normal Density Curve:

Note: Standard normal distribution can be obtained from a given normal distribution by using the following substitution.

xz

Proportion of values for the standard normal distribution can be obtained using the Table B.2 in the appendix B, pages 618-620.

Mean of the Standard Normal Distribution?

Variance of the Standard Normal Distribution?

The normal distribution with parameter values 0 and 1 is called the standard normal distribution. We shall use

the letter z to denote a variable that has this distribution. The corresponding density function is

22

21)( zezf for z

STAT3010 Lecture 1

4

Example: One of the side effects of flooding a lake in northern boreal forest areas (e.g. for a hydro-electric project) is that mercury is leached from the soil, enters the food chain, and eventually contaminates the fish. The concentration in fish will vary among individual fish because of differences in eating patterns, movements around the lake, etc. Suppose that the concentrations of mercury in individual fish follows an approximate normal distribution with a mean of 0.25 ppm and a standard deviation of 0.08 ppm. Fish are safe to eat if the mercury level is below 0.30 ppm. What proportion of fish are safe to eat?

EXPONENTIAL DISTRIBUTIONDefinition

A variable x is said to have an exponential distribution with parameter 0 if the density function for x is

otherwise

xexf

x 0

0)(

STAT3010 Lecture 1

5

Proportion?

Mean of the exponential Distribution?

Variance of the exponential Distribution?

Example: Medical employees at a state hospital have determined that the average time between patient arrivals at the emergency room is exponentially distributed with a mean time of 11 minutes. What is the probability that a patient will arrive within the first 5 minutes?

STAT3010 Lecture 1

6

THE LOGNORMAL DISTRIBUTION

Lognormal distributions are related to normal distributions in exactly the way the name suggests.

Definition:

The density curve:

A nonnegative variable x is said to have a lognormal distribution if ln(x) has a normal distribution with parameters and . It can be shown that the density function of x is

00

02

1

)(

)2/(])[ln( 22

x

xex

xf

x

STAT3010 Lecture 1

7

Mean of the lognormal distribution?

Variance of the lognormal distribution?

Example: Lifetimes of a certain cell are lognormally distributed with parameters 1 day and 5.0 days. Find the probability that a cell lasts longer than four days.

STAT3010 Lecture 1

8

THE WEIBULL DISTRIBUTIONDefinition:

This Weibull distribution is related to the exponential distribution: when 1, the Weibull density function reduces to the exponential density function (with 1 ).

The density curve:

A variable x has a Weibull distribution with parameters and if the density function of x is

00

0

)(

)/(1

x

xex

xf

x

STAT3010 Lecture 1

9

Proportion?

Mean of the Weibull Distribution?

Variance of the Weibull Distribution?

STAT3010 Lecture 1

10

Example: An article suggests that x= cell lifetimes follows a Weibull distribution. Suppose that parameter values are 5.1and 125 . What proportion of cells have a lifetime between 150 and 200.