3rd Edition: Chapter 1

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Chapter 11Sums of Independent R.V.s and Limit
Theorems

• 11.1 Moment-Generating Functions
• 11.2 Sums of Independent R.V.s
• 11.3 Markov and Chebyshev Inequalities
• 11.4 Laws of Large Numbers
• 11.5 Central Limit Theorem

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11.1 Moment Generating Functions

• Definition
• For a random variable X, let
• If MX(t) is defined for all values of t in some
• interval (??, ?), ? gt 0, then MX(t) is called the
  
• moment-generating function of X.
• Discrete R.V.
• Continuous R.V.

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Theorem 11.1
Let X be a R.V. with moment-generating function
MX(t). Then where MX(n)(t) is the nth
derivative of MX(t). Pf
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Corollary
The MacLaurins series of MX(t) is given
by Therefore, E(X n) is the coefficient of t
n/n! in the MacLaurins series representation of
MX(t).
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Example 11.1
Let X be a Bernoulli R.V. with parameter p.
Determine MX(t) and E(X n). Sol
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Example 11.2
Let X be a binomial R.V. with parameter (n, p).
Determine MX(t) and use it to calculate E(X) and
Var(X) . Sol
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Example 11.3
Let X be an exponential R.V. with parameter ?.
Using MX(t) to calculate E(X) and Var(X) . Sol
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Example 11.4
Let X be an exponential R.V. with parameter ?.
Using MX(t) to calculate E(X n), where n is a
positive integer. Sol
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Example 11.5

• Let Z be a standard normal R.V.
• Calculate MZ(t).
• Use part (a) to find MX(t), where X is a normal
  R.V. with mean ? and variance ?2.
• (c) Use part (b) to calculate the mean and the
  variance of X.
• Sol

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Example 11.6

• A positive R.V. X is called lognormal with
  parameter ? and ?2 if lnXN(?, ?2). Let X be a
  lognormal R.V. with parameter ? and ?2.
• For a positive integer r, calculate E(X r).
• Use E(X r) to find Var(X).
• Sol

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Theorem 11.2
Let X and Y be two R.V.s with MX(t) and MY(t).
If for some ? gt 0, MX(t) MY(t) for all values
of t in (??, ?), then X and Y have the same
distribution.
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Example 11.7
Let the moment-generating function of a R.V. X
be Find the PMF of X. Sol
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Example 11.8
Let X be a R.V. with moment-generating
function Find P(0lt Xlt 1). Sol
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11.2 Sum of Independent R.V.s
Theorem 11.3 Let X1, X2, , Xn be independent
R.V.s with moment-generating functions
. Then the moment-generating
function of X1X2Xn is given by Pf
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Theorem 11.4
Let X1, X2, , Xr be independent binomial R.V.s
with parameters (n1, p), (n2, p),, (nr, p),
respectively. Then X1X2Xr is a binomial R.V.
with parameters n1n2 nr and p. Pf
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Theorem 11.5
Let X1, X2, , Xn be independent Poisson R.V.s
with means ?1, ?2,, ?n , respectively. Then
X1X2Xn is a Poisson R.V. with mean ?1 ?2
?n. Pf
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Theorem 11.6
Let X1N(?1, ?12), X2N(?2, ?22), , XnN(?n,
?n2), be independent R.V.s. Then Pf
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Theorem 11.7
Let X1N(?1, ?12), X2N(?2, ?22), , XnN(?n,
?n2), be independent R.V.s. Then for constants
?1, ?2 ,, ?n Pf
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Some Other Useful Results

• Sums of independent geometric R.V.s are negative
  binomial R.V.
• Sums of independent negative binomial R.V.s are
  also negative binomial R.V.
• Sums of independent exponential R.V.s are gamma
  R.V.
• Sums of independent gamma R.V.s are also gamma
  R.V.

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Geometric Negative Binomial
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Moment generating function of Gamma
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Example 11.9

• Suppose that the distribution of students grades
  in a
• probability test is normal, with mean 72 and
  variance 25.
• What is the probability that the average of grade
  of such a probability class with 25 students is
  75 or more?
• If a professor teaches two different sections of
  this course, each containing 25 students, what is
  the probability that the average of one class is
  at least 3 or more than the average of the other
  class?
• Sol

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Example 11.10
Office fire insurance policies by a certain
company have a 1000 deductible. The company has
received 3 claims, independent of each other, for
damages caused by office fire. If reconstruction
expenses for such claims are exponentially
distributed, each with mean 45000. What is the
probability that the total payment for these
claims is less than 120000? Sol
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Example 11.11
Let X1, X2, , Xn be independent standard normal
R.V.s. Then X12X22Xn2, referred to as
chi-squared R.V. with n degrees of freedom, is
gamma with parameters (n/2, ½). An example of
such a gamma R.V. is the error of hitting a
target in n-dimensional Euclidean space when the
error of each coordinate is individually
normally distributed. Pf
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11.3 Markov and Chebyshev Inequalities

• Theorem 11.8

Let X be a nonnegative R.V. then for any t
gt0, Pf
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Example 11.12
A post office, on average, handles 10000 letters
per day. What can be said about the probability
that it will handle (a) at least 15000 letters
tomorrow (b) less than 15000 letters
tomorrow? Sol
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Theorem 11.9 (Chebyshevs Inequality)
If X is a R.V. with expected value ? and variance
? 2 then for any t gt0, Pf
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Chebyshevs Inequality and Sample Mean
Let X1, X2, , Xn be a random sample of size n
from a distribution function F with mean ? and
variance ? 2. Let X be the mean of the
sample. Then converges to the distribution of a
standard normal R.V. That is,
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Example 11.13
For the scores on an achievement test given to a
certain population of students, the expected
value is 500 and the standard deviation is 100.
Let X be the mean of the scores of a random
sample of 10 students. Find a lower bound for
P(460 lt X lt540). Sol
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11.4 Laws of Large Numbers
Theorem 11.10
Let X1, X2, , Xn be a sequence of iid R.V.s with
? E(Xi) and ? 2 Var(Xi) lt ?, i 1, 2, 3, .
Then ?? gt 0,
Definition Let X1, X2, , Xn be a sequence of
R.V.s defined on a sample space. We say that Xn
converges to a R.V. X in probability if, for
each ? gt 0,
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Theorem 11.11 (Strong Law of Large Number)
Let X1, X2, , Xn be an iid sequence of R.V.s
with ? E(Xi), i 1, 2, 3, . Then Pf
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11.5 Central Limit Theorem
Theorem 11.12
Let X1, X2, , Xn be a sequence of iid R.V.s,
each with expectation ? and variance ? 2. Then
the distribution of converges to the
distribution of a standard normal R.V. That is,
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Proof of Central Limit Theorem 11.12
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Theorem 11.13
Let X1, X2, be a sequence of R.V.s with
distribution functions F1, F2, and
moment-generating functions
, respectively. Let X be a R.V. with
distribution function F and moment-generating
function MX(t). If for all values of t, MXn(t)
converges to MX(t), then at the points of
continuity of F, Fn converges to F.
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Example 11.14
If 20 random numbers are selected independently
from the interval (0,1) , what is the probability
that the sum of these numbers is at least 8? Sol