Download Random Variables

Chapter 5
Probability Densities
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Continuous Random
Variables
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Random Variable
For a given sample space S of some
experiment, a random variable is any
rule that associates a number with
each outcome in S .
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Types of Random Variables
A discrete random variable is an rv whose
possible values either constitute a finite
set or else can listed in an infinite
sequence. A random variable is
continuous if its set of possible values
consists of an entire interval on a number
line.
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Random Variables
Represents a possible numerical value
from a random event
Random
Variables
Discrete
Random Variable
Continuous
Random Variable
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Continuous Random Variables
A random variable X is continuous if its
set of possible values is an entire
interval of numbers (If A < B, then any
number x between A and B is possible).
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Continuous Probability Distribution
Let X be a continuous rv. Then a
probability distribution or probability
density function (pdf) of X is a function
f (x) such that for any two numbers a
and b,
P a  X  b    f (x )dx
b
a
The graph of f is the density curve.
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Probability Density Function
For f (x) to be a pdf
1. f (x) > 0 for all values of x.
2.The area of the region between the
graph of f and the x – axis is equal to 1.
y  f ( x)
Area = 1
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Probability Density Function
P (a  X  b )is given by the area of the shaded
region.
y  f ( x)
a
b
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Probability for a Continuous rv
If X is a continuous rv, then for any
number c, P(x = c) = 0. For any two
numbers a and b with a < b,
P (a  X  b )  P (a  X  b )
 P (a  X  b )
 P (a  X  b )
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The Cumulative Distribution Function
The cumulative distribution function,
F(x) for a continuous rv X is defined for
every number x by
F (x )  P  X  x    f ( y )dy
x

For each x, F(x) is the area under the
density curve to the left of x.
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Using F(x) to Compute Probabilities
Let X be a continuous rv with pdf f(x)
and cdf F(x). Then for any number a,
P  X  a   1  F (a)
and for any numbers a and b with a < b,
P a  X  b   F (b )  F (a)
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Obtaining f(x) from F(x)
If X is a continuous rv with pdf f(x)
and cdf F(x), then at every number x
for which the derivativeF ( x ) exists,
F (x )  f (x ).
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Percentiles
Let p be a number between 0 and 1. The
(100p)th percentile of the distribution of a
continuous rv X denoted by  ( p ), is
defined by
p  F  ( p )   
(p)

f ( y )dy
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Median
The median of a continuous distribution,
denoted by  , is the 50th percentile. So 
satisfies 0.5  F (  ). That is, half the area
under the density curve is to the left of  .
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of Continuous Random
Variables
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Expected Value
The expected or mean value of a
continuous rv X with pdf f (x) is
X  E  X  

 x  f ( x)dx

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Expected Value of h(X)
If X is a continuous rv with pdf f(x) and
h(x) is any function of X, then
E  h( x )    h ( X ) 

 h( x)  f ( x)dx

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Variance and Standard Deviation
The variance of continuous rv X with
pdf f(x) and mean  is
2
X

 V ( x) 
 (x  )

2
 f ( x)dx
 E[ X    ]
2
The standard deviation is  X  V ( x).
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Short-cut Formula for Variance
    E ( X )
V (X )  E X
2
2
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Covariance between two discrete
random variables:
σxy =  [x – E(x)] [y – E(y)] f(xy)
where:
xi = possible values of the x discrete random
variable
yj = possible values of the y discrete random
variable
P(xi ,yj) = joint probability of the values of xi and yj
occurring
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Let X and Y be random variables,
The Covariance of X and Y is
 XY  E [(X  X )(Y  Y )
 E (XY )  X Y
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Correlation Coefficient
• The Correlation Coefficient shows
the strength of the linear association
between two variables
σxy
ρ
σx σy
where:
ρ = correlation coefficient (“rho”)
σxy = covariance between x and y
σx = standard deviation of variable x
σy = standard deviation of variable y
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Interpreting the
Correlation Coefficient
• The Correlation Coefficient always falls
between -1 and +1
=0
x and y are not linearly related.
The farther  is from zero, the stronger the linear
relationship:
 = +1
x and y have a perfect positive linear
relationship
 = -1
x and y have a perfect negative linear
relationship
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4.3
Mean and Variance of Linear
Combinations of Random
Variables
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Expected Value of Linear Combinations
E (aX  b)  a  E ( X )  b
This leads to the following:
1. For any constant a,
E (aX )  a  E ( X ).
2. For any constant b,
E ( X  b)  E ( X )  b.
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The Expected Value of
E [ g (X )  h (X )]  E [ g (X )]  E [h (X )]
Also
E [ g (X ,Y )  h (X ,Y )] 
E [ g (X ,Y )]  E [h (X ,Y )]
Finally
E [Y  X ]  E [Y ]  E [X ]
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If X and Y are independent, then
E (XY )  E (X )E (Y )
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• Expected value of the sum of two
discrete random variables:
E(x + y) = E(x) + E(y)
=  x P(x) +  y P(y)
(The expected value of the sum of two
random variables is the sum of the two
expected values)
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Rules of Variance
V (aX
2
 b)   aX b
a
2
2
 X
and  aX b  a   X
This leads to the following:
1.
2.
2
2
2
 aX  a   X ,  aX
2
2
 X b   X
 a  X
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If X and Y are random variables,
then

2
aX bY
 a   b   2ab XY
2
2
X
2
2
Y
If X and Y are Independent
random variables, then

2
aX bY
 a  b 
2
2
X
2
2
Y
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If X and Y are Independent
random variables, then

2
aX bY
 a  b 
2
2
X
2
2
Y
If X1, X2, … Xn are
Independent random variables,
then 2
a X
1
a
2
1
1 a2 X 2 ...an X n
2
X1
a 
2
2
2
X2

 ...  a 
2
n
2
Xn
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Special Continuous
Probability Densites
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Uniform Distribution
A continuous rv X is said to have a
uniform distribution on the interval [A, B]
if the pdf of X is
 1
A x B

f  x; A, B    B  A
 0
otherwise
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The Normal Distribution
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Normal Distributions
A continuous rv X is said to have a
normal distribution with parameters
 and  , where       and
0   , if the pdf of X is
1
 ( x   )2 /(2 2 )
f ( x) 
e
  x  
 2
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Standard Normal Distributions
The normal distribution with parameter
values   0 and   1 is called a
standard normal distribution. The
random variable is denoted by Z. The
pdf is
1
 z2 / 2
f ( z;0,1) 
e
  z  
 2
The cdf is
z
 ( z )  P( Z  z ) 

f ( y;0,1)dy

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Standard Normal Cumulative Areas
Shaded area = (z )
Standard
normal
curve
0
z
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Standard Normal Distribution
Let Z be the standard normal variable.
Find (from table)
a. P( Z  0.85)
Area to the left of 0.85 = 0.8023
b. P(Z > 1.32)
1  P( Z  1.32)  0.0934
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c. P(2.1  Z  1.78)
Find the area to the left of 1.78 then
subtract the area to the left of –2.1.
= P( Z  1.78)  P( Z  2.1)
= 0.9625 – 0.0179
= 0.9446
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z Notation
z will denote the value on the
measurement axis for which the area
under the z curve lies to the right of z .
Shaded area
 P(Z  z )  
0
z
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Ex. Let Z be the standard normal variable. Find z if
a. P(Z < z) = 0.9278.
Look at the table and find an entry
= 0.9278 then read back to find
z = 1.46.
b. P(–z < Z < z) = 0.8132
P(z < Z < –z ) = 2P(0 < Z < z)
= 2[P(z < Z ) – ½]
= 2P(z < Z ) – 1 = 0.8132
P(z < Z ) = 0.9066
z = 1.32
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Nonstandard Normal Distributions
If X has a normal distribution with
mean  and standard deviation  , then
Z
X 

has a standard normal distribution.
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Normal Curve
Approximate percentage of area within
given standard deviations (empirical
rule).
99.7%
95%
68%
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Ex. Let X be a normal random variable
with   80 and   20.
Find P( X  65).
65  80 

P  X  65   P  Z 

20 

 P  Z  .75
= 0.2266
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Ex. A particular rash shown up at an
elementary school. It has been
determined that the length of time that the
rash will last is normally distributed with
  6 days and   1.5 days.
Find the probability that for a student
selected at random, the rash will last for
between 3.75 and 9 days.
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96
 3.75  6
P  3.75  X  9   P 
Z

1.5 
 1.5
 P  1.5  Z  2
= 0.9772 – 0.0668
= 0.9104
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Percentiles of an Arbitrary Normal
Distribution
(100p)th percentile
(100 p)th for 








,

for normal 

standard normal 
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Normal Approximation to the
Binomial Distribution
Let X be a binomial rv based on n trials, each
with probability of success p. If the binomial
probability histogram is not too skewed, X
may be approximated by a normal distribution
with

 np and   npq .
 x  0.5  np 
P( X  x)   



npq


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Ex. At a particular small college the pass rate
of Intermediate Algebra is 72%. If 500
students enroll in a semester determine the
probability that at least 375 students pass.
  np  500(.72)  360
  npq  500(.72)(.28)  10
 375.5  360 
P( X  375)   
  (1.55)
10


= 0.9394
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The Gamma
Distribution
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The Gamma Function
For   0, the gamma function
( ) is defined by

 1  x
( )   x
e dx
0
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Gamma Distribution
A continuous rv X has a gamma
distribution if the pdf is
 1
 1  x / 
x e
x0
 
f ( x; ,  )    ( )

0
otherwise

where the parameters satisfy   0,   0.
The standard gamma distribution has   1.
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Mean and Variance
The mean and variance of a random
variable X having the gamma distribution
f ( x;  ,  ) are
E( X )     V ( X )    
2
2
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Exponential Distribution
A continuous rv X has an exponential
distribution with parameter  if the pdf is
 e x x  0
f ( x;  )  
0

otherwise
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Mean and Variance
The mean and variance of a random
variable X having the exponential
distribution
   
1

   
2
2
1

2
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Probabilities from the Gamma
Distribution
Let X have a exponential distribution
Then the cdf of X is given by
x0
 0
F ( x;  )  
 x
x0
1  e
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Applications of the Exponential
Distribution
Suppose that the number of events
occurring in any time interval of length t
has a Poisson distribution with parameter  t
and that the numbers of occurrences in
nonoverlapping intervals are independent
of one another. Then the distribution of
elapsed time between the occurrences of
two successive events is exponential with
parameter    .
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The Chi-Squared Distribution
Let v be a positive integer. Then a
random variable X is said to have a chisquared distribution with parameter v if
the pdf of X is the gamma density with
  v / 2 and   2. The pdf is
1

(v / 2)1  x / 2
x
e
 v/2
f ( x; v)   2 (v / 2)

0

x0
x0
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
The Chi-Squared Distribution
The parameter v is called the number of
degrees of freedom (df) of X. The
2
symbol  is often used in place of “chisquared.”
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Other Continuous
Distributions
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The Weibull Distribution
A continuous rv X has a Weibull
distribution if the pdf is
   1 ( x /  )
  x e
f ( x;  ,  )   

0

x0
x0
where the parameters satisfy   0,   0.
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Mean and Variance
The mean and variance of a random
variable X having the Weibull
distribution are
2

1
2  
1  

2
2 
     1        1       1    
 
         
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Lognormal Distribution
A nonnegative rv X has a lognormal
distribution if the rv Y = ln(X) has a
normal distribution the resulting pdf has
parameters  and  and is
 1
[ln( x )  ]2 /(2 2 )
e

f ( x;  ,  )   2 x

0

x0
x0
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Mean and Variance
The mean and variance of a variable X
having the lognormal distribution are
E( X )  e
  2 / 2
V (X )  e
2   2
e
2

1
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Beta Distribution
A rv X is said to have a beta distribution
with parameters A, B,   0, and   0
if the pdf of X is
f ( x;  ,  , A, B) 
 1
 1
 1
(   )  x  A   B  x 



 

 B  A ( )  (  )  B  A   B  A 

0
otherwise

x0
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Mean and Variance
The mean and variance of a variable X
having the beta distribution are
  A  ( B  A) 
 
2

 
( B  A) 
2
(   ) (    1)
2
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