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Random Variables
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1
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:

p  x   P X  x 
X

Properties:
p x  0
X
 p x  1
x
3
X
Probability Density Function (PDF):
Continuous R.V.

A continues r.v. is described by a probability density
function fX
Pa  X  b    f  x dx
b
a

Properties:
X
 f  x dx  1

X

f x  0
X
Interpretation: P x  X  x      f  x dx  f  x 
x 

x
4
X
X
Expectation: Discrete R.V.

Definition:
E X    xp  x 
x
X

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)
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Properties of Expectation

Let X be the r.v. and let Y = g(X)
- Hard : EY    yp  y 
y
Y
- Easy : EY    g ( x) p  x 
x
Caution: In general, Eg ( X )  g E X 


X
Properties: If α and β are constants, then:
1) E  
2) EX  
3) EX    
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Variance: Discrete R.V.
Recall: Eg  X    g ( x) p  x 

X
x
Second Moment: Eg  X

2
   x p  x 
2
Variance: var( X )  E X  E X 

X
x
2

 var( X )    X  E X  p  x 
2
X
x
 var( X )  E X   E X 
2
2
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Properties: 1)var( X )  0

2)var(X   )   var( X )
2
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Mean and Variance: Continuous R.V.
 E X    xf  x dx


X
 Eg  X    g  x  f  x dx


 var( X )     ( x  E X ) f  x dx

2
X

2
X

Example: Continuous Uniform r.v.
uniform a  x  b
f x  
otherwise
 0
X
1) for a  x  b, f  x  
2) E  X  
3) var  X  
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X
X
Cumulative Distribution Function (CDF)

Discrete r.v.
 F  x   P X  x    p  x 
X

kx
X
Continuous r.v.
 F  x   P X  x    f  x dx
x
X
dF  x 
 f x 
dx
X
X

Example:
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
X
Mixed Distributions
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Gaussian (Normal) PDF



1
Standard Normal: N 0,1  f  x  
e
2
 x2
2
X
Bell shaped curve:
Expectation and variance: 1) E X  
2) var  X  
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
General Normal:
1
N  ,    f  x  
e
2
X

2
  x   2
2 2
1

e
2 
Expectation and variance: 1) E X  
2) var  X  
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  x   2
2 2