probability_distributions.ppt

Probability Distributions

1. Probability distributions

2. Discrete probability distributions

3. Continuous probability distributions

4. Multidimensional probability distributions

Probability Distributions

Random (stochastic) variables» Experimental measurements are not reproducible in a

deterministic fashion» Each measurement can be viewed a random variable X» Defined on sample space S of an experiment

Probability distribution» Determines probability of particular events» Discrete distributions: random variables are discrete

quantities» Continuous distributions: random variables are

continuous quantities

Cumulative probability distribution function F(x)

)()()()()( aFbFbXaPxXPxF

Discrete Probability Distributions

Random variable X can only assume countably many discrete values: x1, x2, x3, …

Probability density function f(x)

Cumulative distribution function

Properties

otherwise0

if)()( jjj xxxXPpxf

xx xx

jj

j j

pxfxF )()(

jj

bxaj

p

paFbFbXaPj

1

)()()(

Discrete Distribution Example

Matlab function: unicdf(x,n),

>> x = (0:6);

>> y = unidcdf(x,6);

>> stairs(x,y)

Continuous Probability Distributions

Random variable X can assume infinitely many real values


Probability density function

Properties

xdvvfxF )()(

dx

xdFxf

)()(

1)(

)()()()(

dvvf

dvvfaFbFbXaPb

a

Continuous Distribution Example

Probability density function


Probability of events

1)1(75.0)(otherwise0

11)1(75.0)(

1

1

22

dvvdvvfxx

xf

11

1125.075.05.0

10

)(

25.075.05.0)1(75.0)()(

3

3

1

2

x

xxx

x

xF

xxdvvdvvfxFxx

73.025.075.05.095.0)()(

%75.68)1(75.0)5.0()5.0()5.05.0(3

5.0

5.0

2

xxxxFxXP

dvvFFxP

Mean and Variance

Discrete distribution

Continuous distribution

Symmetric distribution» If f(c-x) = f(c+x), then f(x) is symmetric with respect to = c

Transformation of mean and variance» Given random variable X with mean and variance 2

» The standardized random variable Z has zero mean and unity variance

jjj

jjj

xfx

xfx

)()(Variance

)(Mean

22

dxxfx

dxxxf

)()(Variance

)(Mean

22

X

Z

Expectations and Moments

Moments for continuous distributions

Continuous distribution example

dxxfxXEk

dxxfxXEk

XEdxxfxXEdxxxf

kk

kk

)()()]([moment centralth

)()(momentth

)]([)()()()( 222

1

1

1

15

513

312222

1

1

1

14

412

212

2

2.0)75.0()75.0()1(75.0)0()]([

0)75.0()75.0()1(75.0)(

otherwise0

11)1(75.0)(

xxdxxxXE

xxdxxxXE

xxxf

Binomial Distribution

Governs randomness in games of chance, quality inspections, opinion polls, etc.

X = {0,1,2,…,n} = number of times event A occurs in n independent trials

Probability of obtaining A exactly x times in n trials

Mean and variance:

>> binopdf(x,n,p) binomial probability

>> binopdf(0,200,0.02) ans = 0.0176

>> binocdf(x,n,p) binomial cumulative probability

>> 1 - binocdf(100,162,0.5) ans = 0.0010

pqAXPpAXP 1)()(

nn

xnx

xnx

n

x

nxfqpqp

x

nxf

2

1

)!(!

!

2

1)()( 2

1

npqnp 2

Poisson Distribution

Infinitely many possible events Probability distribution function

Limit of binomial distribution as

Mean and variance: 2 = Example

» Probability of a defective screw p = 0.01» Probability of more than 2 defects in a lot of 100 screws?» Binomial distribution: = np = (100)(0.01) = 1» Since p <<1, can use Poisson distribution to approximate solution

• Matlab functions: poisspdf(x,mu), poisscdf(x,mu)

,2,1,0!

)( xex

xfx

constant0 npnp

%03.8)2(1)2(

9197.0!

)()2()2(2

0!2

1!1

1!0

112

0

210

XPXP

eex

xfFXPj j

x

jj

j

Matlab: Poisson-Binomial Comparison Screw example (p=0.01, n=100)

for less than 0–5 defective screws

>> x = (0:5);

>> y = poisscdf(x,1);

>> z = binocdf(x,100,0.01)

Compare Poisson to Binomial with equal , different p and n

Number Binomial Poisson0 0.366032 0.3678791 0.735762 0.7357592 0.920627 0.9196993 0.981626 0.9810124 0.996568 0.996345 0.999465 0.999406

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

0.4

0.5

0.6

0.7

0.8

0.9

1

Binomial n = 50, p = .02Binomial n = 20, p = .05Binomial n = 10, p = .1Binomial n = 5, p = .2Poisson mu=1

Normal (Gaussian) Distribution Probability density function


Standardized normal distribution ( = 0, = 1)

1)(2

1exp

2

1)(

2

dxxfx

xf

dvv

xFx

2

2

1exp

2

1)(

x

xFduezz u

)(2

1)( 2

2

Computing Probabilities

Interval probabilities

Sigma limits

Example» X is a random variable with and 2 = 4

» Use Table A7 in text

abaFbFbXaP )()()(

%7.99)33(

%5.95)22(

%68)(

XP

XP

XP

%797939.0)82.0(2

8.044.2)44.2()44.2(

FXP

Matlab: Normal Distribution

Normal distribution: normpdf(x,mu,sigma)» normpdf(8,10,2) ans = 0.1210» normpdf(9,10,2) ans = 0.1760» normpdf(8,10,4) ans = 0.0880

Normal cumulative distribution: normcdf(x,mu,sigma)» normcdf(8,10,2) ans = 0.1587» normcdf(12,10,2) ans = 0.8413

Inverse normal cumulative distribution: norminv(p,mu,sigma)» norminv([0.025 0.975],10,2) ans = 6.0801 13.9199

Random number from normal distribution: normrnd(mu,sigma,v)» normrnd(10,2,[1 5]) ans = 9.1349 6.6688

10.2507 10.5754 7.7071

Matlab: Normal Distribution Example

The temperature of a bioreactor follows a normal distribution with an average temperature of 30oC and a standard deviation of 1oC. What percentage of the reactor operating time will the temperature be within +/-0.5oC of the average?

Calculate probability at 29.5oC and 30.5oC, then calculate the difference:» p=normcdf([29.5 30.5],30,1)p = [0.3085 0.6915]» p(2) – p(1)0.3829

The reactor temperature will be within +/- 0.5oC of the average ~68% of the operating time

25 26 27 28 29 30 31 32 33 34 350

0.1

0.2

0.3

0.4

Temperature

Multidimensional Probability Distributions

Consider two random variable X and Y

Two-dimensional cumulative distribution function

Discrete distributions

Continuous distribution

Marginal distributions

),(),( yYxXPyxF

i j

jixx yy

ji yxfyxfyxFi j

1),(),(),(

2

2

1

1

),(),(

1***)*,(***)*,(),(

2211

b

a

b

a

y x

dxdyyxfbYabXaP

dydxyxfdydxyxfyxF

dyyxfxf

yxfYxXPxfj

j

),()(Continuous

),()arbitrary ,()(Discrete

1

1

Independent Random Variables

Basic property

Addition and multiplication of means

Addition of variance

Covariance

)()(),()()(),( 2121 yfxfyxfyFxFyxF

)()()()(only st variableIndependen

)()()()(General

2121

2121

nn

nn

XEXEXEXXXE

XEXEXEXXXE

XY

YEXEYXE

ZYXYX

2

)()()(

},{:},{:},{:

2`2

2`1

221

2211

22

21

2

)()()(st variableIndependen

)()()(

YEXEXYE

YEXEXYEXY

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