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

Lecture slides on continuous random variables and probability distributions, including the uniform and normal distributions, calculating probabilities, assessing normality, and approximating the binomial distribution using the normal distribution. It also covers the concept of continuous probability density functions and their relationship to probability.

Typology: Slides

2011/2012

Uploaded on 07/03/2012

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Lecture: Continuous pdfs
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Download Lecture Slides: Continuous Random Variables & Probability Distributions and more Slides Mathematical Modeling and Simulation in PDF only on Docsity!

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 (11.5x11.8) = (Base)(Height)
  • = (11.8 - 11.5)(1) = 0.
  1. 0 1

1

  1. 5 11. 5

1 1

 

dc

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

  1. Can Be Used to Approximate Discrete Probability

Distributions Example: Binomial

  1. Basis for Classical Statistical Inference

Normal Distribution

  1. ‘Bell-Shaped’ &

Symmetrical

  1. Mean, Median,

Mode Are Equal

  1. 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