Mathematics for Computer Science
MIT 6.042J/18.062J
Expectations:
sums & products
Copyright © Albert R. Meyer, 2007. All rights reserved.
May 11, 2007
lec 13F.1
Linearity of Expectation
E [ aR ] = a E [ R ]
E [ R + S] = E [ R ] + E [ S]
Follows by rearranging terms in
def of E[·], for example:
Copyright © Albert R. Meyer, 2007. All rights reserved.
May 11, 2007
lec 13F.2
Linearity of Expectation
X
E [ aR ] ::=
=
ar Pr f aR = ar g
ar 2 range( aR )
X
ar Pr f R = r g
r 2 range( R )
X
=a
r Pr f R = r g
r 2 range( R )
= a E [R]
Copyright © Albert R. Meyer, 2007. All rights reserved.
May 11, 2007
lec 13F.3
Conditional Expectation
The expectation of R given event A:
E [ R j A ] ::=
X
r Pr f R = r j A g
r 2 range( R )
Copyright © Albert R. Meyer, 2007. All rights reserved.
May 11, 2007
lec 13F.4
Conditional Expectation
example: D the roll of a fair die.
Pr{D=1} =  =Pr{D=6} = 1/6
E[D] = (1+2++6)·(1/6) = 3.5
But given that the roll > 3,
what is the expectation?
E [D j D > 3]?
Copyright © Albert R. Meyer, 2007. All rights reserved.
May 11, 2007
lec 13F.5
Conditional Expectation
Pr f D = 1 j D > 3g = 0;
likewise for D = 2; 3:
Pr f D = 4 j D > 3g = 1=3;
likewise for D = 5; 6
Copyright © Albert R. Meyer, 2007. All rights reserved.
May 11, 2007
lec 13F.6
Conditional Expectation
E [D j D ¸ 4]
=
X6
i ¢Pr f D = i j D ¸ 4g
i= 1
= 1 ¢0 + 2 ¢0 + 3 ¢0+
1
1
1
4 ¢3 + 5 ¢3 + 6 ¢3 = 5
Copyright © Albert R. Meyer, 2007. All rights reserved.
May 11, 2007
lec 13F.7
Law of Total Expectation
Like Law of Total Probability, allows
reasoning by cases.
Copyright © Albert R. Meyer, 2007. All rights reserved.
May 11, 2007
lec 13F.8
Team Problems
Problems
1&2
Copyright © Albert R. Meyer, 2007. All rights reserved.
May 11, 2007
lec 13F.9