63
Chapter 3. Discrete Random Variables and Probability Distributions
STAT 155Suppose that independent Bernoulli trials, each with probabilitypof being a success, are performed until a total number ofrsuccesses occurs, whereris a specified positive integer.The random variableX= the number of failures that precedetherth success, is called anegative binomial random variablewith parameterspandr.We writeX⇠NB(r, p).Its pmf is:nb(x;r, p) =P(X=x) =✓x+r1r1◆pr(1p)x,x= 0,1,2,· · ·And its meanand varianceareE(X) =r(1p)pV(X) =r(1p)p2ExampleWhen a fisherman catches a fish, if it is young with a probabilityof 0.2, the fisherman returns the fish to the water. On the other hand, anadult fish will be kept. Suppose the fisherman sets a goal of 10 adult fish.(a) What is the expected number of young fish caught by the fishermanbefore the 10th adult fish is caught?(b) What is the expected number of all fish the fisherman needs to catchin order to reach the goal?
64
Chapter 3. Discrete Random Variables and Probability Distributions
STAT 155
Exercise 3.75Suppose thatp=P(male birth) =.5. A couple wishes tohave exactly two female children in their family.They will have childrenuntil this condition is fulfilled.(a) What is the probability that the family hasxmale children?(b) What is the probability that the family has four children?(c) What is the probability that the family has at most four children?(d) How many male children would you expect this family to have? Howmany children would you expect this family to have?
65
Chapter 3. Discrete Random Variables and Probability Distributions
STAT 155
Geometric rv as a special case of Negative Binomial rvSuppose we perform the independent Bernoulli trials until onesuccess occurs. The random variableX= the number of failures, is ageometric random variablewith parameterp.That is,X⇠Geometric(p)is equivalent toX⇠NB(1, p).We have the geometric pmfs,g(x;p) =P(X=x) =p(1p)x,x= 0,1,2,· · ·And the meansand variancesareE(X) =1ppV(X) =1pp2ExampleWhen a fisherman catches a fish, if it is young with a probabilityof 0.2, the fisherman returns the fish to the water. On the other hand, anadult fish will be kept.(a) What is the expected number of fish caught by the fisherman until thefirst adult fish is caught?(b) What is the probability that the fifth fish caught is the first young fish?(c) If the fisherman catches five fish, what is the probability that there areexactly one young fish?
66
Chapter 3. Discrete Random Variables and Probability Distributions
STAT 155
3.6 The Poisson Probability Distribution
A discrete random variable
X
is said to have a
Poisson
distribution
with parameter
μ
(
μ >
0)
if the pmf of
X
is,
p
(
x
;
μ
) =
P
(
X
=
x
) =
e

μ
μ
x
x
!
,
x
= 0
,
1
,
2
, . . .
The mean
and variance
of the Poisson rv
X
are
E
(
X
) =
μ
V
(
X
) =
μ
Poisson distribution deals with counting the number of times
an event occurs in a given interval (time, space, volume, etc.).