Chapter 7 Properties of Expectation

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Chapter 7 Properties of Expectation

• Expectation of Sums of Random Variables
• Covariance, Variance of Sums, and Correlations
• Conditional Expectation
• Conditional Expectation and Prediction
• Moment Generating Function
• Additional Properties of Normal Random Variables

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Properties of Expectation

• Booles Inequality
• Let A1, , An denote events then

By Induction
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So Y is equal to 1 if at least one of the Ai
occurs
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For any non-negative, integer-values random
variables X, define
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Example Mean of a Negative Binomial Random
Variables

• If independent trails, having a constant
  probability p of being successes, are performed,
  determine the expected number of trials required
  to achieve a total or r successes.
• Let X1 be the number of trails required to obtain
  the first success.
• X2 be the additional number of trails required
  to obtain the second success.
• Then the number of trails needed to achieve a
  total of r successes X is
• X X1 X2 Xr
• The r.v. Xi is a geometric random variable with
  parameter p
• EXi 1/p, i1,2,,r
• EX EX1 EX2 EXr r/p
• Let nr Average number of trials needed to
  achieve a total of r successes
• nr (1nr-1)p (1nr)(1-p)
• pnr pnr-1 1
• n1 1/p
• nr nr-1 1/p
• nr r/p

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Mean of Hypergeometric Random Variable

• If n balls are randomly selected from an urn
  containing N balls of which m are white, find the
  expected number of white balls selected.
• Let
• Then the number of white balls selected is

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Example Suppose that a jar contains 2N cards,
two of them marked 1, two marked 2, two marked 3
etc. Draw out m cards at random. What is the
expected number of pairs that still remain in the
jar? Let
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Coupon-Collecting Problem Suppose that there are
N types of coupons (a) Find the expected number
of different types of coupons that are contained
in a set of n coupons. (b) Find the expected
number of coupons needed to obtain a complete set
of coupons. Sol (a)
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(b) Let Yi, i 0,1,,N-1, be the number of
additional coupons needed after i distinct types
have been collected in order to obtain another
distinct type and Y Y1 Y2 Yn-1 is the
total of coupons collected before a complete
set is obtained. example 111313214111332245 Y01
, Y13, Y23, Y32, Y49, Yi is a geometric
r.v. with parameter (N-i)/N, thus
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Covariance, Variance of Sums, and Correlations
Covariance between X and Y, Cov(X,Y) E(X
EX)(Y EY) EXY EXEY If X and
Y are independent, then Cov(X,Y)0 However,
Cov(X,Y)0 X,Y are independent.

• Cov(X,Y) Cov(Y,X)
• Cov(X,X) Var(X)
• Cov(aX,Y)aCov(X,Y)

If X1, , Xn are pairwise independent, then
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Example Compute the variance of a binomial r.v.
X with parameter n and p X X1 X2
Xn Xi are independent Bernoulli r.v. EXi 1p
0(1-p) p Var(Xi) EXi2
(EXi)2 EXi - (EXi)2 (since
EXi EXi2) p - p2 Thus,
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Compute the variance of X, the number of people
that select their own hats. Let
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The correlation coefficient of X and Y
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Example Suppose X and Y are the number of heads
and tails respectively when a coin is tossed
twice. What are Cov(X,Y) and ?(X,Y)?
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Example Let IA and IB be indicator variables for
events A and B.
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Expected Number of Runs. Suppose that a sequence
of n 1s and m 0s is randomly permuted so that
each of the (mn)!/m!n! possible arrangement is
equally likely. Any consecutive string of 1s is
said to constitute a run of 1s, for instance, if
n6, m4 1110110010 then there are 3 runs of
1s and 3 runs of 0s.
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Conditional Expectation
The conditional probability mass function of X,
given that Yy is
Conditional expectation of X, given Yy is
Conditional probability density of X, given that
Yy is
Conditional expectation of X, given that Yy is
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Computing expectations by conditioning
Expectation of a random number of random
variables. X1, X2, , Xn are i.i.d. r.v. and N
is a r.v. independent of Xis. Then
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Example An urn contains a white and b black
balls. One ball at a time is randomly withdrawn
until the first white ball is drawn. Find the
expected number of black balls that are
withdrawn. Let X denote the number of black balls
withdrawn, and EX Ma,b Let
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Example Let U1, U2, be a sequence of
independent uniform (0,1) random variables. Find
EN, such that
of uniform (0,1) r.v. we need add until their
sum exceeds x.
and set
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Conditional Variance of X given Yy
Var(XY) EX-E(XY)2Y EX2Y
(EXY)2 So EVar(XY) EX2 E( E(XY)
)2 Also VarEXY EEXY2
(EEXY)2 EEXY2
EX2 EVar(XY) Var(EXY) EX2 EX2
Var(X)
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Example Suppose that by any time t, the number
of people that have arrived at a train station is
a Poisson random variable with mean ?t. If the
initial train arrives at the station at a time
that is uniformly distributed over (0,T), what is
the mean and variance of the number of passengers
that enter the train? N(t) of arrivals by
time t Y the time at which the train arrives.
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Example Show that E(x-a)2 is minimized at
aEX.
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Example Suppose X is a continuous random
variable with density f. Show that EX-a is
minimized when a is equal to the median of F.
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Moment Generating Function
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Binomial distribution with parameter n, p
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Poisson Distribution with mean ?
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Exponential Distribution with parameter ?
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Normal Distribution ZN(0,1)
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If X and Y are independent r.v., then
If X and Y are independent binomial random
variables with parameters (n,p) and (m,p), then
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If X and Y are Normal r.v. with parameters
(µ1,s12) and (µ2,s22), respectively,
For any n random variables X1, X2, , Xn, the
joint moment generating function