Chapter 3Some Special Distributions

Math 6203Fall 2009

Instructor: Ayona Chatterjee

3.1 The Binomial and Related Distributions

Bernoulli Distribution

• A Bernoulli experiment is a random experiment in which the outcome can be classified in one of two mutually exclusive and exhaustive ways.– Example: defective/non-defective, success/failure.

• A sequence of independent Bernoulli trials has a fixed probability of success p.

• Let X be a random variable associated with a Bernoulli trial. X = 1 implies a success. X = 0 implies a failure.

• The pmf of X can be written as : p(x)=px (1-p)1-x x=0,1

• In a sequence of n Bernoulli trials, we are often interested in a total of X number of successes.

Binomial Experiment

• Independent and identical n trials.• Probability of success p is fixed for each trial.• Only two possible outcomes for each trial.• Number of trails are fixed.

Binomial Distribution

• The random variable X which counts the number of success of a Binomial experiment is said to have a Binomial distribution with parameters n and p and its pmf is given by:

elsewhere

nxppx

nxp

xnx

0

,...2,1,0)1()(

Theorem

on.distributi )b( binomial has YThen .Let

. m ..., 2, 1, ifor p),b(n, binomial a has

such that variablesrandomt independen be ,...,Let

11

21

,pnXY

X

XXX

m

ii

m

ii

i

m

Negative Binomial Distribution

• Consider a sequence of independent repetitions of a random experiment with constant probability p of success. Let the random variable Y denote the total number of failures in this sequence before the rth success that is, Y +r is equal to the number of trials required to produce exactly r successes. The pmf of Y is called a Negative Binomial distribution

• Thus the probability of getting r-1 successes in the first y+r-1 trials and getting the rth success in the (y+r)th trial gives the pmf of Y as

elsewhere

yppr

ryyp

yr

Y

0

,....2,1,0)1(1

1)(

Geometric Distribution

• The special case of r = 1 in the negative binomial, that is finding the first success in y trials gives the geometric distribution.

• Thus we can re-write • P(Y)=p qy-1 for y = 1, 2, 3, …. • Lets find mean and variance for the Geometric

Distribution.

Multinomial Distribution

• Define the random variable Xi to be equal to the number of outcomes that are elements of Ci , i = 1, 2, … k-1. Here C1, C2, … Ck are k mutually exhaustive and exclusive outcomes of the experiment. The experiment is repeated n number of times. The multinomial distribution is

nxxx

pppxxx

n

k

xk

xx

k

k

121

2121

..

....!!..!

!21

Trinomial Distribution

• Let n= 3 in the multinomial distribution and we let X1 = X and X2= Y, then n –X-Y = X3 we have a trinomial distribution with the joint pmf of X and Y given as

1

)!(!!

!),(

321

)(321

ppp

nyx

pppyxnyx

nyxp yxnyx

3.2 The Poisson Distribution

• A random variable that has a pmf of the form p(x) as given below is said to have a Poisson distribution with parameter m.

elsewhere

xm

emxp

mx

0

,....2,1,0!)(

Poisson Postulates• Let g(x,w) denote the probability of x changes

in each interval of length w. • Let the symbol o(h) represent any function

such that • The postulates are– g(1,h)=λh+o(h), where λ is a positive constant and

h > 0.– and – The number of changes in nonoverlapping

intervals are independent.

.0]/)([lim0

hhoh

2

)(),(x

hohxg

Note

• The number of changes in X in an interval of length w has a Poisson distribution with parameter m = wλ

Theorem

.parameteron with distributiPoisson a has Then

.mparameter on with distributiPoisson a has

such that variablesrandomt independen be ,...,Let

11

i

21

n

ii

n

ii

i

m

mXY

X

XXX

3.3 The Gamma, Chi and Beta Distributions

• The gamma function of α can be written as

)!1()(

1)1(

)(0

1

dyey y

The Gamma Distribution

• A random variable X that has a pdf of the form below is said to have a gamma distribution with parameters α and β.

elsewhere

xexxf

x

0

0)(

1)(

/1

Exponential Distribution

• The gamma distribution is used to model wait times.

• W has a gamma distribution with α = k and β= 1/ λ. If W is the waiting time until the first change, that is k = 1, the pdf of W is the exponential distribution with parameter λ and its density is given as

elsewhere

wewg

w

0

0)(

Chi-Square Distribution

• A special case of the Gamma distribution with α=r/2 and β=2 gives the Chi-Square distribution. Here r is a positive integer called the degrees of freedom.

2/1,)21()(

0

02)2/(

1)(

2/

2/12/2/

tttM

elsewhere

xexrxf

r

xrr

Theorem

on.distributi )r( has YThen .X YLet

on.distributi )(r a has X that n, ..., 1, ifor Suppose,

. variablesrandomt independen be ,XLet *

Corollary

on.distributi ),( has YThen .X YLet

on.distributi ),( a has X that n, ..., 1, ifor Suppose,

. variablesrandomt independen be ,XLet *

2

22

)(

by given isit and exists )E(X

then r/2- k If on.distributi )( a have XLet *

1i

2

1i

i2

i

1

1i

1i

ii

1

k

2

n

i

n

i

n

n

i

n

i

n

k

k

X

X

r

kr

XE

r

Beta Distribution

• A random variable X is said to have a beta distribution with parameters α and β if its density is given as follows

elsewhere

yeyyg

y

0

0)(

1)(

1

The Normal Distribution

• We say a random variable X has a normal distribution if its pdf is given as below. The parameters μ and σ2 are the mean and variance of X respectively. We write X has N(μ,σ2).

x

xxf

2

2

1exp

2

1)(

The mgf

• The moment generating function for X~N(μ,σ2) is

22

2

1exp)( tttM

Theorems

on.distributi )/,N( a has XThen

.nXLet on.distributi ),N(

common a with variablesrandom iid be XLet -Corollary*

).,N( is Y ofon distributi Then the constants. are s'

a YLet on.distributi ),N( has X n, ..., 1, i

for such that variablesrandomt independen be XLet *

).1( is /)-(XV vrailerandom then the

,0),,N( is X variablerandom theIf*

2

1

1-2

1

1

22

1

1i

2i

1

222

22

n

X

X

aaa

X

X

n

ii

n

n

iii

n

iiii

n

iiii

n

3.6 t and F-Distributions

The t-distribution

• Let W be a random variable with N(0,1) and let V denote a random variable with Chi-square distribution with r degrees of freedom. Then

• Has a t-distribution with pdf

rV

WT

/

t

rtrr

rtTg

r 2/)1(2 )/1(

1

)2/(

]2/)1[()(

The F-distribution

• Consider two independent chi-square variables each with degrees of freedom r1 and r2.

• Let F = (U/r1)/(V/r2)• The variable F has a F-distribution with

parameters r1 and r2 and its pdf is

f

rwr

w

rr

rrrrfg rr

rr

0

)/1(

)(

)2/()2/(

)/](2/)[()( 2/)(

21

1)2/(

21

2/2121

21

11

Student’s Theorem

freedom. of degrees 1-non with distributi-student t

a has nS/

-XT variablerandom The (d)

on.distributi )1( a has /1)-(n (c)

t.independen are and (b)

on.distributi )/,( a has (a)

,

)(1

1 and

1

, variablesrandom theDefine

. varianceand mean on with distribuit Normal

a havingeach variablesrandom iid be ,,XLet

222

2

2

1

22

1

2

21

nS

SX

nNX

Then

XXn

SXn

X

XX

n

ii

n

ii

n