Continuous random variables, Slides of Probability and Statistics

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CRV :

Continuous

Random

Variables

PDF : Probability Density Function

Expectation

Variance

Uniform Distribution

• Uniform random variable on (a,b) X : U(a,b)

0,otherwise

, a x b

b a

f x

2 2 3 3

2 2

2 2 2

2 2

a b b a b a b a

a b dx b a

x

a b E X

a b b a b a

dx b a

E X x

b

a

b

a





Exponential Distribution

• Exponential random variable X :

Memoryless property

0, otherwise

( ) e , x 0

• x

=

= 

 f x 

2

2 2 2

0



^ =



 −

E X

E X x e dx

x

Exp ( )

P ( X  x + y |X  x)=P(X  y)

MemoryLess Property

Exponential Distribution

Exponential Distribution

x

f ( x )

Time Between Successive Arrivals (mins.)

Example : Al’s Full-Service Pump

P ( x < 2) = 1 - 2.71828-2/3^ = 1 - .5134 =.

Normal and standard normal

distributions:

The Normal Distribution:

as mathematical function (pdf)

( ) 2

1





 

− −

x

f x e

Note constants:

e=2.

This is a bell shaped curve

with different centers and

spreads depending on  a

nd 

The Normal PDF

2 ( ) 2

e dx

x

It’s a probability function, so no matter what the values of 

and , must integrate to 1!

Normal curve specification:

No matter what  and  are, the area between - and 

+ is about 68%; the area between - 2  and +2 is ab

out 95%; and the area between - 3  and +3 is about

99.7%. Almost all values fall within 3 standard deviatio

ns.

68 - 95 - 99.7 Rule

68% of th

e data

95% of the data

99.7% of the data