Lecture 15

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Example: Rope 4 meters long - break can occur at any point. The variable (X) is the point were the break occurs

What is P(X=2) ?

What is P(0<X<2)?

P(X=2) has no meaning for continuous variable

For continuous variables the Probability distribution is described by a Probability Density Function

What does the Probability Density Function look like for this situation?

Probability Density Function for the rope problem

f(x) = 0.25

Where 0<x<4

From the probability density function we can determine probabilities

for example,

What is P(1<x<2) ?

This probability is given by the area under the prob. density function curve [or the integral of f(x)]

E{X} is integral of x times f(x) wrt to dx

Variance of x is integral of square of (x - E{x}) times f(x) wrt dx

Example: Rope problem

f(x) = 0.25

What is E{X}?

What is s2{X}?

The rope problem is a general form of distribution which is referred to as the Uniform Distribution.

The Uniform Distribution has a rectangular shape which is only applicable over a specified interval

If a Uniform Distribution is applicable over the interval from c to d show that

f(x) = 1/(d-c)

and

Expected value = (c+d)/2

 

Lecture 16

 

 

Exponential Distribution

Used to study DURATION phenomena

Mirror image of the Poisson distribution

Exponential - used to study TIME (or distance) between occurrences

Poisson - used to study NUMBER OF OCCURRENCES in a given time period

Same problem - different variable

Exponential Probability Density Function

f(x) = [exp(-x/q)]/ q

where x>0

n = q s = q

 

Exponential

In other ways, q , is the average time between arrival.

For Poisson, l, is the average number of arrivals.

For a given problem, l = 1/q

 

Determining Exponential Probabilities

P(x<a) = intergral of f(x) from x=0 to x=a

= 1 - exp(-a/q)

The book gives, P(x>a) = exp(-a/q)

 

Exponential Example

Example

The number of cars arriving at a garage can be modeled by a Poisson distribution with an average time of 1.25 seconds between cars

Find the probability that the time between arrivals is a) less than 2 seconds, b) between 1 and 2 seconds, c) greater than 1.5 seconds

a) P(x<2sec) = 1 - exp(- 2/1.25) = 1 - 0.20 = 0.80

b) P(1<x<2) = P(x<2) - P(x<1) =

0.80 - (1 - exp(-1/1.25) = 0.80 -0.45 = 0.35

c) P(x>1.5) = exp (-1.5/1.25) = 0.30

Lecture 17

Normal distribution can be used to model many real life problems

Some examples of normal distribution

Wear pattern on steps

Height distribution

Strength of Concrete

Location of Falling Balls (Boston Science Museum)

Normal Probability Density Function

 

Parameters

Two parameters - m and s

E{X} = m

s2{X} = s2

Notation

N(m,s2)

This notation means we have

a normal distribution

mean of m

variance of s2

A standard normal distribution is the normal distribution for a standardized variable

Mean? Variance?

Standard Normal, N(0,1)

Z, is used to represent the standard normal variable

Any normally distributed variable, X, can be converted to Z by standardization

Z = (X-m)/s

The probability distribution of any normal variable can be obtained from the standard normal table

The percentage of material passing the #200 sieve for aggregate from L&T Quarry is normally distributed with mean of 10% and variance of 4 (%)2.

The specifications require that the percent passing the #200 sieve be between 7% and 14%.

FIND

a) The probability that a randomly selected sample will have too much fines

b) The probability that the sample will have too little fines

c) The probability that the sample will meet specs