Download Random Variables

Bridges
RANDOM VARIABLES.docx
RANDOM VARIABLES
FINITE
 Probability Mass Function (pmf):
f X ( x)  p ( X  x)
 Cumulative Distribution Function (cdf):
FX ( x)  P( X  x)
 Mean of X or Expected Value of X :
 X  E ( x)   x  f X ( x)
all x
𝑉(𝑋) = ∑𝑎𝑙𝑙 𝑥(𝑥 − 𝜇𝑋 )2 ∙ 𝑓𝑋 (𝑥)
 Variance of X :
 If X is a Bernoulli Random Variable, then its distribution is binomial. The Expected Value
of a binomial distribution is: E ( x)  n  p
CONTINUOUS
f X ( x )  P ( a  X  b)
 Probability Density Function (pdf):
 Cumulative Distribution Function (cdf): FX ( x)  P ( X  x)

 Mean of X or Expected Value of X :
 X  E ( x)    x  f ( x)dx
 Variance of X :
𝑉(𝑋) = ∫−∞(𝑥 − 𝜇𝑋 )2 ∙ 𝑓𝑋 (𝑥) 𝑑𝑥
∞
 If X is an Exponential Random Variable, then
if x  0
0
f X ( x)  
( x / )
if x  0
(1/  )  e
0
FX ( x)  
( x / )
1  e
if x  0
if x  0
 If X is an Uniform Random Variable on [a, b], then
0 if x  a


f X ( x)  1 /(b  a) if a  x  b


0 if x  b
0
if x  a


FX ( x)  ( x  a) /(b  a) if a  x  b


1
if x  b
 If X is a Normal Random Variable, then
𝑓𝑋 (𝑥 ) =
1
𝜎𝑋 ∙√2𝜋
𝑥−𝜇
∙
2
−0.5( 𝜎 𝑋 )
𝑋
𝑒