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Random Variables

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Random Variables An assignment of a value (number) to every possible

outcome. Mathematically: A function from the sample space Ω to

the real numbers.− discrete or continuous values.

Can have several random variables defined on the same sample space.

Notation:− random variable X− Numerical value x

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Probability Mass Function (PMF): Discrete R.V. Probability distribution of X Notation:

Properties:

xXxpX

P

1

0

x

X

X

xp

xp

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Probability Density Function (PDF): Continuous R.V. A continues r.v. is described by a probability density

function fX

Properties:

Interpretation:

dxxfbXab

aXP

0

1

xf

dxxf

X

X

xfdxxfxXxX

x

xX

P

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Expectation: Discrete R.V. Definition:

Interpretation:− Center of gravity of PMF− Average in large number of repetitions of the

experiment

Example: Uniform on 0,1, 2,…, n. Find E(X)

x

XxxpXE

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Properties of Expectation Let X be the r.v. and let Y = g(X)

Caution: In general,

Properties: If α and β are constants, then:

xX

yY

xpxgY

yypY

)(E:Easy-

E:Hard-

XgXg E)(E

X

X

E 3)

E 2)

E 1)

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Variance: Discrete R.V. Recall:

Second Moment:

Variance:

Properties:

x

XxpxgXg )(E

x

XxpxXg 22E

22

2

2

EE)ar(

E)ar(

EE)ar(

XXXv

xpXXXv

XXXv

xX

)ar()ar()2

0)ar()12 XvXv

Xv

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Mean and Variance: Continuous R.V.

Example: Continuous Uniform r.v.

dxxfXxX

dxxfxgXgdxxxfX

XX

XX

22 )E()var(

E E

otherwise

bxauniformxf

X 0

X

X

xfbxaX

var)3

E )2

,for )1

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Cumulative Distribution Function (CDF) Discrete r.v.

Continuous r.v.

Example:

xk

XXxpxXxF P

dx

xdFxf

dxxfxXxF

X

X

x

XX

P

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Mixed Distributions

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Gaussian (Normal) PDF Standard Normal: Bell shaped curve: Expectation and variance:

General Normal:

Expectation and variance:

2

2

2

11,0

x

XexfN

X

X

var)2

E )1

2

2

2

2

22

2 2

1

2

1,

xx

XeexfN

X

X

var)2

E )1

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