Mathematics for Computer Science
MIT 6.042J/18.062J
Expected Number
of Heads
Albert R Meyer,
May 8, 2013
lec 12F.1
Expected #Heads
n independent flips of a
coin with bias p for Heads.
How many Heads expected?
E[# Heads]
= E[B n,p ]
Albert R Meyer,
May 8, 2013
lec 12F.3
Expected #Heads
n independent flips of a
coin with bias p for Heads.
How many Heads expected?
ænæ k
n-k
E[B n,p ] ::= æ k æ æp (1 - p)
k=0 ækæ
n
Albert R Meyer,
May 8, 2013
lec 12F.4
Expected #Heads
n independent flips of a
coin with bias p for Heads.
How many Heads expected?
ænæ k n-k
E[B n,p ] ::= æ k æ æp q
k=0 ækæ
n
Albert R Meyer,
May 8, 2013
lec 12F.5
Expected #Heads
Binomial theorem and
differentiating gives a
closed formula
Albert R Meyer,
May 8, 2013
lec 12F.6
Binomial Expectation
(x + y)
n
=æ
take  / x :
n x  y
n 1
n
k=0
ænæ k n-k
æ æx y
ækæ
 n  k n k
1
 k   x y
x k 0  k 
Albert R Meyer,
n
May 8, 2013
lec 12F.7
Binomial Expectation
ænæ k n-k
æ
æ
E B n,p ::= æ k æ æp q
æ æ
k=0 ækæ
n
(
n x+y
)
n-1
ænæ k n-k
= æ k æ æx y
x k=0 ækæ
1
Albert R Meyer,
n
May 8, 2013
lec 12F.8
Binomial Expectation
ænæ k n-k
E æB n,p æ::= æ k æ æp q
æ æ
k=0 ækæ
n
( )
n p+q
n-1
ænæ k n-k
= æ k æ æp q
p k=0 ækæ
1
Albert R Meyer,
n
May 8, 2013
lec 12F.9
Binomial Expectation
 n  k n k
E Bn,p  ::  k   p q
 
k 0  k 
n
n
ænæ k n-k
= æ k æ æp q
p k=0 ækæ
1
Albert R Meyer,
n
May 8, 2013
lec 12F.10
Binomial Expectation
 n  k n k
E Bn,p  ::  k   p q
 
k 0  k 
n
n
1 æ æ
= E B n,p
p æ æ
æ
æ
np = E B n,p
æ æ
Albert R Meyer,
May 8, 2013
lec 12F.11