Download Random Variable

Introduction to
Probability & Statistics
Random Variables
Random Variables
A Random Variable is a function that
associates a real number with each element in a
sample space.
Ex:
Toss of a die
X = # dots on top face of die
= 1, 2, 3, 4, 5, 6
Random Variables
A Random Variable is a function that
associates a real number with each element in a
sample space.
Ex:
Flip of a coin
X=

0 , heads
1 , tails
Random Variables
A Random Variable is a function that
associates a real number with each element in a
sample space.
Ex:
Flip 3 coins

X=
0
1
2
3
if
if
if
if
TTT
HTT, THT, TTH
HHT, HTH, THH
HHH
Random Variables
A Random Variable is a function that
associates a real number with each element in a
sample space.
Ex:
X = lifetime of a light bulb
X = [0, )
Distributions
Let X = number of dots on top face of a die
when thrown
p(x) = Prob{X=x}
x
p(x)
1
2
3
4
5
6
1/ 1/ 1/ 1/ 1/ 1/
6
6
6
6
6
6
Cumulative
Let F(x) = Pr{X < x}
x
1
2
3
4
5
6
p(x)
1/
F(x)
1/ 2/ 3/ 4/ 5/ 6/
6
6
6
6
6
6
1/ 1/ 1/ 1/ 1/
6
6
6
6
6
6
Complementary Cumulative
Let F(x) = 1 - F(x) = Pr{X > x}
x
1
2
3
4
5
6
p(x)
1/ 1/ 1/ 1/ 1/ 1/
6
6
6
6
6
6
F(x)
1/ 2/ 3/ 4/ 5/ 6/
6
6
6
6
6
6
F(x)
5/ 4/ 3/ 2/ 1/ 0/
6
6
6
6
6
6
Discrete Univariate
 Binomial
 Discrete
Uniform (Die)
 Hypergeometric
 Poisson
 Bernoulli
 Geometric
 Negative
Binomial
Binomial Distribution
n=8, p=.5
0.5
0.5
0.4
0.4
0.3
0.3
P(x)
P(x)
n=5, p=.3
0.2
0.1
0.2
0.1
0.0
0.0
0
1
2
3
4
5
0
1
x
4
0.4
0.3
P(x)
0.3
0.2
0.1
0.2
0.1
0.0
3
4
5
x
6
7
8
4
2
2
1
0
0.0
0
5
n=20, p=.5
0.5
0.4
P(x)
3
x
n=4, p=.8
0.5
2
x
Continuous Distribution
f(x)
A
x
a
1. f(x) > 0
b
c
,
d
all x
d
2.  f ( x)dx  1
a
c
3. P(A) = Pr{a < x < b} =  f ( x)dx
a
4. Pr{X=a} =  f ( x)dx  0
a
b
Continuous Univariate





Normal
Uniform
Exponential
Weibull
LogNormal









Beta
T-distribution
Chi-square
F-distribution
Maxwell
Raleigh
Triangular
Generalized Gamma
H-function
Normal Distribution
1
f ( x) 
2
F
IJ
G
H
K
e
1 X 

2 
  
2

65%
95%
99.7%



Std. Normal Transformation
f(z)
Standard Normal
X 
Z


1
f ( z) 
e
2
1
 z2
2

N(0,1)

Example
 Suppose
a resistor has specifications of 100 +
10 ohms. R = actual resistance of a resistor
and R
N(100,5). What is the probability a
resistor taken at random is out of spec?

LSL
USL
x
0
100
0
Example Cont.
LSL
USL
x
0
Pr{in spec}
100
0
= Pr{90 < x < 110}
90  100 x   110  100


 Pr 




5
5

= Pr(-2 < z < 2)
Example Cont.
LSL
USL
x
0
Pr{in spec}
100
0
= Pr(-2 < z < 2)
= [F(2) - F(-2)]
= (.9773 - .0228) = .9545
Pr{out of spec} = 1 - Pr{in spec}
= 1 - .9545
= 0.0455
Example
Suppose the distribution of student grades for
university are approximately normally
distributed with a mean of 3.0 and a standard
deviation of 0.3. What percentage of students
will graduate magna or summa cum laude?
x
3.0
.5
Example Cont.
x
3.0
.5
Pr{magna or summa} = Pr{X > 3.5}}
 Pr X    3.5  3.0 
0.3 
 
= Pr(z > 1.67)
= 0.5 - 0.4525
= 0.0475
Example
 Suppose
we wish to relax the criteria so that
10% of the student body graduates magna or
summa cum laude.
0.1
x
3.0
?
Example
 Suppose
we wish to relax the criteria so that
10% of the student body graduates magna or
summa cum laude.
0.1
x
3.0
0.1 = Pr{Z > z}
z = 1.282
?
Example
0.1
x
3.0
But
X
Z

?
x =  + z
= 3.0 + 0.3 x 1.282
= 3.3846
Exponential Distribution
Density
f ( x )  e
Cumulative
F ( x)  1  e
Mean
1/
Variance
1/2
 x
,x>0
 x
1.0
Density
=1
0.5
0.0
0
0.5
1
1.5
Time to Fail
2
2.5
3
Exponential Distribution
Density
f ( x )  e
Cumulative
F ( x)  1  e
Mean
1/
Variance
1/2
 x
,x>0
 x
2.0
Density
1.5
=1
=2
1.0
0.5
0.0
0
0.5
1
1.5
Time to Fail
2
2.5
3
Example
Let X = lifetime of a machine where the life is
governed by the exponential distribution.
determine the probability that the machine
fails within a given time period a.
f ( x)  e  x , x > 0,  > 0
Example
Exponential Life
2.0
1.8
1.6
F ( a )  Pr{X  a}
a
 x
e
 0  dx
e
 x a
0
1 e 
a
Density
f ( x)   e
 x
1.4
1.2
f(x)
1.0
0.8
0.6
0.4
0.2
0.0
0
0.5
1
a
1.5
2
Time to Fail
2.5
3
Complementary
Exponential Life
2.0
1.8
1.6
Density
Suppose we wish to know
the probability that the
machine will last at
least a hrs?
1.4
1.2
0.8
0.6
F ( a )  Pr{X  a}
0.4

0.0
 a e
a
 e 
 x
f(x)
1.0
0.2
0
dx
0.5
1
a
1.5
2
Time to Fail
2.5
3
Example
Suppose for the same exponential distribution, we
know the probability that the machine will last at
least a more hrs given that it has already lasted c hrs.
a
c
c+a
Pr{X > a + c | X > c}
= Pr{X > a + c  X > c} / Pr{X > c}
= Pr{X > a + c} / Pr{X > c}
e   (ca )
 a
e
  c 
e