Chapter 8
Special Discrete
Distributions
Binomial Distribution B(n,p)
• Each trial results in one of two
mutually exclusive outcomes.
(success/failure)
• There are a fixed number of trials
• Outcomes of different trials are
independent
• The probability that a trial results in
success is the same for all trials
• The binomial random variable x is
defined as the number of successes out
of the fixed number
Are these binomial distributions?
1) Toss a coin 10 times and count the
number of heads
Yes
2) Deal 10 cards from a shuffled deck
and count the number of red cards
No, probability does not remain constant
3) Two parents with genes for O and A
blood types and count the number of
children with blood type O
No, no fixed number
Toss a 3 coins and count the number of
heads
Find the discrete probability distribution
X
P(x)
0
1
2
3
.125
.375
.375
.125
Out of 3 coins that are tossed, what is
the probability of getting exactly 2 heads?
Binomial Formula:
n  k
n k
P (x  k )    p 1  p 
k
 
Where:
n 
 n C k
k 
Out of 3 coins that are tossed,
what is the probability of
getting exactly 2 heads?
3 2
1
P (x  2)   0.5 0.5  .375
2
The number of inaccurate gauges in a
group of four is a binomial random
variable. If the probability of a defect
is 0.1, what is the probability that only
1 is defective?
 4 1
3
P (x  1)   0.1 0.9  .2916
1 
More than 1 is defective?
P (x  1)  1  (P (0)  P (1))  .0523
Calculator
• Binomialpdf(n,p,x) – this
calculates the probability of a
single binomial P(x = k)
• Binomialcdf(n,p,x) – this
calculates the cumulative
probabilities from P(0) to P(k)
A genetic trait of one family
manifests itself in 25% of the
offspring. If eight offspring are
randomly selected, find the
probability that the trait will
appear in exactly three of them.
P (X  3)  binomialpdf (8,.25,3)  .2076
At least 5?
P (X  5)  1  binomialcdf (8,.25,4)  .0273
In a certain county, 30% of the
voters are Republicans. If ten
voters are selected at random,
find the probability that no more
than six of them will be
Republicans.
P(x < 6) = binomcdf(10,.3,6) = .9894
Binomial formulas for mean
and standard deviation
 x  np
 x  np 1  p 
In a certain county, 30% of the
voters are Republicans. How
many Republicans would you expect
in ten randomly selected voters?
What is the standard deviation for
this distribution?
 x  10(.3)  3 Republicans
x  10(.3)(.7)  1.45 Republicans
•In L1 – seq(x,x,0,10)
•In L2 – binompdf(10, .1 ,L1)
•Sketch histogram on board
What happened to the
shape of the distribution
as the probability of
success increased?
As the probability of success
increases, the shape changes from
being skewed right to symmetrical
at p =.5 to skewed left.
•Calculate the mean and standard
deviations for each of the
probabilities
What do you notice?
As the probability of success increase,
•the means increase.
•the standard deviations increase to p = .5, then
decrease. Their values are also symmetrical.
Geometric Distributions:
• There are two mutually exclusive
outcomes
• Each trial is independent of the
others
• The probability of success
remains constant for each trial.
• The random variable x is the
number of trials UNTIL the
FIRST success occurs.
Differences between binomial
& geometric distributions
• The difference between
binomial and geometric
properties is that there is
NOT a fixed number of
trials in geometric
distributions!
Other differences:
•Binomial random variables
start with 0 while geometric
random variables start with 1
•Binomial distributions are
finite, while geometric
distributions are infinite
Geometric Formulas:
P (x )  p 1  p 
x 1
1
x 
p
1p
x 
2
p
Count the number of boys in a
family of four children.
Binomial:
X
0
1
2
3
4
Count children until first son
is born
Geometric:
X
1
2
3
4
. . .
What is the probability that
the first son is the fourth
child born?
P (X  4)  geometricpdf (.5,4)  .0625
What is the probability
that the first son is born is
at most four children?
P (X  4)  geometriccdf (.5,4)  .9375
A real estate agent shows a house to
prospective buyers. The probability that
the house will be sold to the person is
35%. What is the probability that the
agent will sell the house to the third
person she shows it to?
P (x  3)  geometricpdf (.35,3)  .1479
How many prospective buyers does she
expect to show the house to before
someone buys the house?
1
x 
 2.86 buyers
.35
•In L1 – input numbers 1-20
•In L2 – geometpdf(.1,L1)
•Sketch
•Find the means & standard
deviations
What do you see?
•Geometric distributions are skewed
right and become more strongly
skewed right as the probability of
success increases
•Mean & standard deviation of the
distributions decrease as the
probability of success increase
Poisson Distributions
This distribution deals with the
probabilities of rare events that
occur infrequently in space, time,
distance, area, etc.
Examples:
• The number of accidents that occur
per month at a given intersection
• The number of tardies per
semester for a given student
• The number of runs per inning in a
baseball game
Properties:
• The occurrence of a success in any
interval is independent of that in any
other interval
• The probability that a success will
occur in any interval is the same for
all intervals of equal size and is
proportional to the size of the
interval
• We observe a discrete number of
events in a continuous (fixed)
interval.
Formulas:
X = number of rare events per unit of time, space, etc.
l = mean value of X (Greek letter lambda)
P (X ) 
x  l
x  l
l e
x
x!
l
The number of accidents in an office
building during a four-week period
averages 2. What is the probability
there will be one accident in the next
four-week period?
21  e 2
P (X  1) 
1!
 .2707
What is the probability that there
will be more than two accidents in
the next four-week period?
P (X  2)  1  (P (0)  ...  P (2))  .3233
8:00 of
untilcalls
8:30to
is aa 30
minute
period.
TheFrom
number
police
department
From 8:00
is apm
60 minute
period.
between
8 pmuntil
and9:00
8:30
on Friday
averages 3.5.
Since the period is doubled, you must
•What
is the
the mean
probability
calls
double
amountof
of no
calls
to during
keep
thisP(X
period?
it proportional! =.0302
= 0) = poissonpdf(3.5,0)
•What is Be
the sure
probability
of
no
calls
to adjust l!
between 8 pm and 9 pm on Friday night?
P(X = 0) = poissonpdf(7,0) =.0009
•What is the mean and standard deviation
of the number of calls between 10 pm and
 = 14 &  = 3.742
midnight on Friday night?
Examine the histograms of the Poisson
distributions
–
What happens
to
What happens to
l = 2 What
the
happens tol = 4
the shape?
means?
the standard
deviations?
l= 6
As l increases
• The distributions become
more symmetrical
• The means increase
• The standard deviations
increase