Introduction to Probability

1
Introduction to Probability StatisticsExpecta
tions
2
Expectations

• Mean

3
Example
Consider the discrete uniform die example

• ?? EX 1(1/6) 2(1/6) 3(1/6)
• 4(1/6) 5(1/6) 6(1/6)
• 3.5

4
Expected Life
For a producted governed by an exponential life
distribution, the expected life of the product
is given by
?
2.0
x
?
?
?
?

x
E
x
e
dx
?
1.8
1.6
1.4
0
1.2
?
x
?
f
t )
e
(x
?
?
1.0
Density
0.8
0.6
0.4
0.2
X
0.0
0
0.5
1
1.5
2
2.5
3
5
Expected Life
For a producted governed by an exponential life
distribution, the expected life of the product
is given by
?
2.0
x
?
?
?
?

x
E
x
e
dx
?
1.8
1.6
1.4
0
1.2
?
x
?
f
t )
e
(x
?
?
1.0
Density
0.8
0.6
0.4
0.2
X
0.0
0
0.5
1
1.5
2
2.5
3
6
Expected Life
For a producted governed by an exponential life
distribution, the expected life of the product
is given by
?
2.0
x
?
?
?
?

x
E
x
e
dx
?
1.8
1.6
1.4
0
1.2
?
x
?
f
t )
e
(x
?
?
1.0
Density
0.8
0.6
0.4
0.2
X
0.0
0
0.5
1
1.5
2
2.5
3
7
Expected Life
For a producted governed by an exponential life
distribution, the expected life of the product
is given by
?
2.0
x
?
?
?
?

x
E
x
e
dx
?
1.8
1.6
1.4
0
1.2
?
x
?
f
t )
e
(x
?
?
1.0
Density
0.8
0.6
0.4
0.2
X
0.0
0
0.5
1
1.5
2
2.5
3
1/?
8
Variance
9
Example

• Consider the discrete uniform die example

?2 E(X-?)2 (1-3.5)2(1/6) (2-3.5)2(1/6)
(3-3.5)2(1/6) (4-3.5)2(1/6) (5-3.5)2(1/6)
(6-3.5)2(1/6) 2.92
10
Property
11
Property
12
Property
13
Example

• Consider the discrete uniform die example

?2 EX2 - ?2 12(1/6) 22(1/6) 32(1/6)
42(1/6) 52(1/6) 62(1/6) - 3.52
91/6 - 3.52 2.92
14
Exponential Example
For a producted governed by an exponential life
distribution, the expected life of the product
is given by
2.0
1.8
1.6
1.4
1.2
?
x
?
f
t )
e
(x
?
?
1.0
Density
0.8
0.6
0.4
0.2
X
0.0
0
0.5
1
1.5
2
2.5
3
0.5
1
1/?
15
Exponential Example
For a producted governed by an exponential life
distribution, the expected life of the product
is given by
2.0
1.8
1.6
1.4
1.2
?
x
?
f
t )
e
(x
?
?
1.0
Density
0.8
0.6
0.4
0.2
X
0.0
0
0.5
1
1.5
2
2.5
3
1/?
16
Exponential Example
For a producted governed by an exponential life
distribution, the expected life of the product
is given by
2.0
1.8
1.6
1.4
1.2
?
x
?
f
t )
e
(x
?
?
1.0
Density
0.8
0.6
0.4
0.2
X
0.0
0
0.5
1
1.5
2
2.5
3
1/?

17
Properties of Expectations

• 1. Ec c
• 2. EaX b aEX b
• 3. ?2(ax b) a2?2
• 4. Eg(x)
• g(x) Eg(x)
• X
• (x-?)2
• e-tx

18
Properties of Expectations

• 1. Ec c
• 2. EaX b aEX b
• 3. ?2(ax b) a2?2
• 4. Eg(x)
• g(x) Eg(x)
• X ?
• (x-?)2 ?2
• e-tx ?(t)

19
Property Derviation

• Prove the property
• Eaxb aEx b

20
Property Derivation
21
Property Derivation
m
1
22
Property Derivation
m
1
am b1 aEx b
23
Class Problem

• Total monthly production costs for a casting
  foundry are given by
• TC 100,000 50X
• where X is the number of castings made during a
  particular month. Past data indicates that X is
  a random variable which is governed by the normal
  distribution with mean 10,000 and variance 500.
  What is the distribution governing Total Cost?

24
Class Problem

• Soln
• TC 100,000 50X
• is a linear transformation on a normal
• TC Normal(mTC, s2TC)

25
Class Problem

• Using property Eaxb aExb
• mTC E100,000 50X
• 100,000 50EX
• 100,000 50(10,000)
• 600,000

26
Class Problem

• Using property s2(axb) a2s2(x)
• s2TC s2(100,000 50X)
• 502 s2(X)
• 502 (500)
• 1,250,000

27
Class Problem

• TC 100,000 50 X
• but,
• X N(100,000 , 500)
• TC N(600,000 , 1,250,000)
• N(600000 , 1118)