Lecture Slides: Continuous Random Variables & Probability Distributions, Slides of Mathematical Modeling and Simulation

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Lecture Slides

on

Modeling and Simulation

Lecture: Continuous pdfs

Continuous Random Variables

• Describe the Uniform and Normal Random Variables
• Calculate Probabilities for Continuous Random Variables
• Assess Normality for a Distribution
• Approximate the Binomial Distribution Using the Normal

Distribution

Continuous Random Variable Probability

P c x d f x dx c

d (   )  (^)  ( )

f(x)

x c d

Probability is Area Under Curve!

Continuous Probability Distribution

Uniform Normal Exponential Other

Continuous Random Variable Probability

Example on Uniform Distribution

Suppose You’re production manager of a soft

drink bottling company.

You believe that when a machine is set to

dispense 12 oz., it really dispenses 11.5 to 12.

oz. inclusive.

Suppose the amount dispensed has a uniform

distribution.

What is the probability that less than 11.8 oz. is

dispensed?

SODA

Uniform Distribution Solution

• P (11.5  x  11.8) = (Base)(Height)
• = (11.8 - 11.5)(1) = 0.

1. 0 1

1

12. 5 11. 5

1 1

 



 d  c

f ( x )

x

% How to generate uniform dis. % The domain is generated x = -1:0.1:11; % now the pdf for x values pdf = unifpdf(x, 0, 10); cdf = unifcdf(x, 0, 10); subplot(1,2,1),plot(x,pdf) title('pdf') xlabel('X'), ylabel('f(x)') axis([-1 11 0 0.2]) axis square subplot(1,2,2),plot(x,cdf) title('cdf') xlabel('X'), ylabel('f(x)') axis([-1 11 0 1.1]) axis square

MATLAB Program to Generate Uniform pdf & cdf

Averages of Continuous Random Variables

The average value of a continuous random variable in an interval [a, b] is

Where, f(x) is the probability density function (pdf) for x. The normalization condition is

The average and variance value of any function of g(x) with this pdf are



 b a

b a x f x dx

E x xdF x

( )

( ) ( )

var{ g }  E ( g^2 ) E ( g )^2

 (^ ) ^1  ()





f x dx F

E g g x f x dx

b

 a

( ) ( ) ( )

Importance of Normal Distribution

1. Describes Many Random Processes or Continuous

Phenomena

2. Can Be Used to Approximate Discrete Probability

Distributions Example: Binomial

3. Basis for Classical Statistical Inference

Normal Distribution

1. ‘Bell-Shaped’ &

Symmetrical

2. Mean, Median,

Mode Are Equal

3. Random Variable

Has Infinite Range

Mean, Median, Mode

x

f(x)

Pdf and cdf

Cumulative Probability func (cdf) is

Probability Density Function (pdf) is

F x x dx

x   

  

 (^)    0

2

2 2

exp ( ) 2

( )^1 

  

2

exp



  

x

f x

Effect of Varying Parameters ( & )

x

f(x)

A C

B

Normal distributions differ by mean & standard deviation.

x

f(x)

A C

B

Infinite Number of Tables

X

f(X)

Normal distributions differ by mean & standard deviation.

Each distribution would require its own table.

That’s an infinite number!

Infinite Numbers of Tables