continuous probability distributions continuous random variables & probability distributions

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Continuous Probability Distributions

Continuous Random Variables &Probability Distributions

Dr. Jerrell T. Stracener, SAE Fellow

Leadership in Engineering

EMIS 7370/5370 STAT 5340 : PROBABILITY AND STATISTICS FOR SCIENTISTS AND ENGINEERS

Systems Engineering ProgramDepartment of Engineering Management, Information and Systems

Stracener_EMIS 7370/STAT 5340_Sum 08_06.05.08


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•Definition - A random variable is a mathematical function that associates a number with every possible outcome in the sample space S.

• Definition - If a sample space contains an infinite number of possibilities equal to the number of points on a line segment, it is called a continuous sample space and a random variable defined over this space is called a continuous random variable.

• Notation - Capital letters, usually or , are used to denote random variables. Corresponding lower case letters, x or y, are used to denote particular values of the random variables or .

X Y

YX

Random Variable


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For many continuous random variables or (probabilityfunctions) there exists a function f, defined for allreal numbers x, from which P(A) can for any eventA S, be obtained by integration:

Given a probability function P() which may berepresented in the form of

A

dxxfAP

areaA

dxxfAP

Continuous Random Variable


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in terms of some function f, the function f is calledthe probability density function of the probabilityfunction P or of the random variable , and the probability function P is specified by the probability density function f.

X



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Probabilities of various events may be obtained from the probability density function as follows:

Let A = {x|a < x < b}

Then

P(A) = P(a < X < b)

A

dxxf

b

a

dxxf



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Therefore = area under the density function curvebetween x = a and x = b.

f(x)

x

Area = P(a < x <b)

a b0 0

)(AP



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The function f(x) is a probability density function for the continuous random variable X, defined over the set of real numbers R, if

1. f(x) 0 for all x R.

2.

3. P(a < X < b) =

.1dx)x(f

b

a

.dx)x(f

Probability Density Function


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The cumulative probability distribution function, F(x), of a continuous random variable with density function f(x) is given by

Note:

x

.dt)t(f)xX(P)x(F

xFdx

df(x)

X

Probability Distribution Function


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Probability Density and Distribution Functions

f(x) = Probability Density Function

x

Area = P(x1 < <x2)

F(x) = Probability Distribution Function

x

F(x2)

F(x1)

x2x1

P(x1 < <x2) = F(x2) - F(x1)

1

cumulative area

x2x1

X

X


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• Mean or Expected Value

• Remark

Interpretation of the mean or expected value:The average value of in the long run.

dxxfx XEμ

X

Mean & Standard Deviationof a Continuous Random Variable X


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•Variance of X:

•

•Standard Deviation of :

dx f(x) μ)-(xσXVar 22

22 μXEXVar

22 μxfx dx

XVarσ X

Mean & Standard Deviationof a Continuous Random Variable X


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If a and b are constants and if = E is the meanand 2 = Var is the variance of the randomvariable , respectively, then

and

baμbaXE

XVarabaXVar 2

X)(X

)(X

Rules


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If Y = g(X) is a function of a continuous random variable , then

dxxfxgxgEμY

X

Rules


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If the probability density function of X is

)(xf)1(2 x

0

for 0 < x < 1

elsewhere

then find

(a) and

(b) P(X>0.4)

(c) the value of x* for which P(X<x*)=0.90

Example


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First, plot f(x):

0

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 1

x

f(x)

Example


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Find the mean and standard deviation of X,

1

0

)()( dxxxfXE

dxxxdxxx 1

0

21

0

][2)1(2

3

1

2

12

322

1

0

32 xx

3

1

3

21

Example Solution


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222 )()( XEXVar

21

0

2

3

1)(

dxxfx

9

1

432

9

1)1(2

1

0

431

0

2

xx

dxxx

9

1

12

2

3

1

4

1

3

12

2

Example Solution


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and the standard deviation is

18

1

12

2

3

1

3

1

4

2

3

1

236.018

1

Example Solution


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(b)

(c)

2

00

2

22 )()()(

xx

dxxdxxfxXPxFxx

)4.0X(P1)4.0X(P

36.064.01

4.04.0*21 2

1.32or 0.68 x*

9.0*)(*)(2*)(*)( 2

therefore

xxxPxXP

Example Solution

for 0<x<1

Since 1.32>1, so 0.68x*


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Uniform Distribution


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Probability Density Function

elsewhere , 0

0afor b, x afor , 1

)( abxf

a b0

f(x)

x

1/(b-a)



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Probability Distribution Function

b x afor ab

ax )()(

xXPxF

a b0

F(x)

x

1

a for x 0

b for x 1



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• Mean

= (a+b)/2

• Standard Deviation

12

ab



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Example – Uniform Distribution


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Example Solution – Uniform Distribution

continuous probability distributions continuous random variables & probability distributions

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