moment generation function

Use of moment generating functions

DefinitionLet X denote a random variable with probability density function f(x) if continuous (probability mass function p(x) if discrete)ThenmX(t) = the moment generating function of X

The distribution of a random variable X is described by either The density function f(x) if X continuous (probability mass function p(x) if X discrete), orThe cumulative distribution function F(x), orThe moment generating function mX(t)

Properties mX(0) = 1

Let X be a random variable with moment generating function mX(t). Let Y = bX + a Then mY(t) = mbX + a(t) = E(e [bX + a]t) = eatmX (bt) Let X and Y be two independent random variables with moment generating function mX(t) and mY(t) . Then mX+Y(t) = mX (t) mY (t)

Let X and Y be two random variables with moment generating function mX(t) and mY(t) and two distribution functions FX(x) and FY(y) respectively. Let mX (t) = mY (t) then FX(x) = FY(x). This ensures that the distribution of a random variable can be identified by its moment generating function

M. G. F.s - Continuous distributions

Name

Moment generating function MX(t)

Continuous Uniform

EQ \F(ebt-eat,[b-a]t)

Exponential

for t <

Gamma

for t <

2

d.f.

EQ \B\bc\[(\F(1,1-2t)) /2 for t < 1/2

Normal

et+(1/2)t22

_889966744.unknown

_889966930.unknown

M. G. F.s - Discrete distributions

Name

Moment generating function MX(t)

Discrete Uniform

EQ \F(et,N) \F(etN-1,et-1)

Bernoulli

q + pet

Binomial

(q + pet)N

Geometric

EQ \F(pet,1-qet)

Negative Binomial

EQ \B\bc\[(\F(pet,1-qet)) k

Poisson

e(et-1)

Moment generating function of the gamma distributionwhere

usingor

then

Moment generating function of the Standard Normal distributionwherethus

We will use

Note:Also

Note:Also

Equating coefficients of tk, we get

Using of moment generating functions to find the distribution of functions of Random Variables

ExampleSuppose that X has a normal distribution with mean m and standard deviation s.Find the distribution of Y = aX + bSolution:= the moment generating function of the normal distribution with mean am + b and variance a2s2.

Thus Z has a standard normal distribution .Special Case: the z transformationThus Y = aX + b has a normal distribution with mean am + b and variance a2s2.

ExampleSuppose that X and Y are independent each having a normal distribution with means mX and mY , standard deviations sX and sY Find the distribution of S = X + YSolution:Now

or= the moment generating function of the normal distribution with mean mX + mY and variance

Thus Y = X + Y has a normal distribution with mean mX + mY and variance

ExampleSuppose that X and Y are independent each having a normal distribution with means mX and mY , standard deviations sX and sY Find the distribution of L = aX + bYSolution:Now

or= the moment generating function of the normal distribution with mean amX + bmY and variance

Thus Y = aX + bY has a normal distribution with mean amX + BmY and variance

Special Case:Thus Y = X - Y has a normal distribution with mean mX - mY and variance

a = +1 and b = -1.

Example (Extension to n independent RVs)Suppose that X1, X2, , Xn are independent each having a normal distribution with means mi, standard deviations si (for i = 1, 2, , n)Find the distribution of L = a1X1 + a1X2 + + anXnSolution:Now(for i = 1, 2, , n)

or= the moment generating function of the normal distribution with mean and variance

Thus Y = a1X1 + + anXn has a normal distribution with mean a1m1 + + anmn and variance

In this case X1, X2, , Xn is a sample from a normal distribution with mean m, and standard deviations s, andSpecial case:

Thusand variance

has a normal distribution with mean

If x1, x2, , xn is a sample from a normal distribution with mean m, and standard deviations s, thenSummaryand variance

has a normal distribution with mean

Population

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Sheet2

Sheet3

If x1, x2, , xn is a sample from a distribution with mean m, and standard deviations s, then if n is large The Central Limit theoremand variance

has a normal distribution with mean

We will use the following fact: Let m1(t), m2(t), denote a sequence of moment generating functions corresponding to the sequence of distribution functions:F1(x) , F2(x), Let m(t) be a moment generating function corresponding to the distribution function F(x) then ifProof: (use moment generating functions)then

Let x1, x2, denote a sequence of independent random variables coming from a distribution with moment generating function m(t) and distribution function F(x).Let Sn = x1 + x2 + + xn then

Is the moment generating function of the standard normal distributionThus the limiting distribution of z is the standard normal distributionQ.E.D.