Distribución de Variables Aleatorias Continuas, Diapositivas de Probabilidad

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Continuous Random

Variables

Random Variable

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

Probability distribution

The function f(x) is a probability

density function, for the

continuous random variable X,

defined over the set of real

numbers R, if

 

 

   

x R

P a X b f x dx

f x dx

f x

b

a







 

for all

Cumulative distribution

The cumulative distribution F(x) of

a continuous random variable X

with density function f(x) is

               

F x P X x f t dt x

x

for

Percentile

In a random variable X, the value

of x up until which the specified

probability () is accumulated.

F(x) = and solve for x.

Percentile

Percentile 0.5 = Median = 2nd

Quartile.

Percentile 0.25 = 1st Quartile.

Percentile 0.75 = 3rd Quartile.

Interquartile Range = 3rd Quartile – 1st

Quartile.

Critical Value  = Percentile (1 – )

Expected Value

Let X be a random variable with

probability distribution f(x). The

mean or expected value of the

random variable g(X) is

      





 

 E g x  g x f x dx g x



Variance

Let X be a random variable with

probability distribution f(x) and

mean . The variance of X is

       

 

2 2

2 2 2



  





 

E X

E X x f x dx

Chebyshev´s Theorem

The probability that any random

variable X will assume a value

within k standard deviations of the

mean is at least 1-1/k

2

. That is,

  2

1

1

k

P   k   X    k   

Continuous Uniform

Distribution

The density function of the

continuous uniform random

variable X on the interval [A, B] is:

 

 







 





elsewhere

A X B

B A

f x A B

0

1

; ,

Normal Distribution

The density function of the normal

random variable X, with mean 

and variance 

2 , is

 

 

     

 

n x e x

x

,

2

1

; ,

2

2

2 



 

 

Normal Distribution

-6 -4 -2 0 2 4 6

Normal Distribution

Exponential Distribution

The continuous random variable X

has an exponential distribution,

with parameter , if its density

function is given by











 



elsewhere

e x f x

x

0

0

(^1) 



Exponential Distribution

The mean and the variance of the

Exponential Distribution are

The cdf of an exponential

distribution is:

2 2

      x x

and

 

e x

x

F x

x

