1. Introduction to Probability & Statistics Expectations

2. Expectations    Mean:   E X xdF x [ ] ( )   xp x discrete ( ) ,   E X
xdF x [ ] ( )    xp x
discrete ( ) ,      xf x dx
continuous ( ) ,

3. Example Consider the discrete uniform die example: x 1 2 3 4 5 6
p(x) /6 1/6 1/6 1/6 1/6 1/6
 = E[X] = 1(1/6) + 2(1/6) + 3(1/6)
+ 4(1/6) + 5(1/6) + 6(1/6)
= 3.5

4. Expected Life   E [ x]  x  e dx   x e dx    ( ) 2  1 
For a producted governed by an exponential life distribution,
the expected life of the product is given by 
2.0 E [ x]   x  e   x dx
1.8
1.6
1.4
1.2 f (x
t )   e   x
1.0      x e dx 2 1
Density
0.8
0.6
0.4
0.2
0.0 X
0.5 1
1.5 2
2.5 3    ( ) 2
1/  1 

5. Variance      ( ) x dF     E x [( ) ] =     ( ) x p   2    ( ) x dF   2   E x [( )
] =   2    ( ) x p   2      ( ) x f dx

6. Example Consider the discrete uniform die example: x 1 2 3 4 5 6
p(x) /6 1/6 1/6 1/6 1/6 1/6
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

7. Property        ( ) x f dx    ( ) x f dx     x f dx xf (2      ( ) x f dx     ( ) x f dx 2      x f dx xf 2 ( ) 

8. Property        ( ) x f dx    ( ) x f dx     x f dx xf (2      ( ) x f dx     ( ) x f dx 2      x f dx xf 2 ( )     E X [ ] 2    E X [ ] 2 

9. Example Consider the discrete uniform die example: x 1 2 3 4 5 6
p(x) /6 1/6 1/6 1/6 1/6 1/6
2 = E[X2] - 2 = 12(1/6) + 22(1/6) + 32(1/6)
+ 42(1/6) + 52(1/6) + 62(1/6)
= 91/
= 2.92

10. Exponential Example       E X [ ]   1   x e dx ( )   x e
For a producted governed by an exponential life distribution,
the expected life of the product is given by   2   E X [ ]
2.0
1.8   2 1    x e dx ( )
1.6
1.4
1.2 f (x
t )   e   x
1.0
Density
0.8      x e dx 3 1
0.6  1 2 
0.4
0.2
0.0 X
0.5 1
1.5 2
2.5 3    ( ) 3
1/ 1 2   1 2  =

11. Properties of Expectations
1. E[c] = c
2. E[aX + b] = aE[X] + b
3. 2(ax + b) = a22
4. E[g(x)] =
g(x) E[g(x)] X
(x-)2
e-tx g x dF ( ) 

12. 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 What is the distribution governing Total Cost?

13. Class Problem Soln: TC = 100,000 + 50X
is a linear transformation on a normal
TC ~ Normal(mTC, s2TC)

14. Class Problem Using property E[ax+b] = aE[x]+b mTC = E[100,000 + 50X]
= 100, (10,000)
= 600,000

15. Class Problem Using property s2(ax+b) = a2s2(x)
s2TC = s2(100, X)
= 502 s2(X)
= 502 (500)
= 1,250,000

16. Class Problem TC = 100,000 + 50 X but, X ~ N(100,000 , 500)
TC ~ N(600,000 , 1,250,000)
~ N( , 1118)