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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