Some Discrete Probability
Distributions
By: Prof. Gevelyn B. Itao
Probability and Statistics
Discrete Uniform Distribution
 If the random variable X assumes the values x1,
x2,…,xk, with equal probabilities, then the discrete
uniform distribution is given by
Probability and Statistics
Discrete Uniform Distribution
Example 5.1: When a light bulb is selected at random from a box
that contains a 40-watt bulb, a 60-watt bulb, a 75-watt bulb, and
a 100-watt bulb, each element of the sample space
S = {40, 60, 75, 100} occurs with probability 1/4.
Therefore, we have a uniform distribution, with
Probability and Statistics
Discrete Uniform Distribution
Example 5.2: When a die is tossed, each element of the sample
space S
S = {1,2,3,4,5,6} occurs with probability 1/6.
Therefore, we have a uniform distribution, with
Probability and Statistics
Theorem 5.1
The mean and variance of the discrete uniform distribution
f (x; k) are
Probability and Statistics
Theorem 5.1
Example 5.3: referring to example 5.2, compute the mean and
variance:
Probability and Statistics
Bernoulli Process
 An experiment that consists of repeated trials, each with two
possible outcomes that may be labeled success or failure.
Properties of Bernoulli Process
1. The experiment consists of n repeated trials.
2. Each trial results in 2 possible outcomes only that may be
classified as a success or failure.
3. The probability of success, denoted by p, remains constant
from trial to trial.
4. The repeated trials are independent.
Probability and Statistics
Binomial random variable
 The number X of successes in n Bernoulli trials
Binomial distribution
 The probability distribution of the discrete binomial random
variable.
 A Bernoulli trial can result in a success with probability p and a
failure probability q = 1 – p. Then the probability distribution
of the binomial random variable X, the number of successes in
n independent trials, is
Probability and Statistics
Binomial distribution
Example 5.4: The probability that a certain kind of component
will survive a given shock test is 3/4. Find the probability that
exactly 2 of the next 4 components tested survive.
Probability and Statistics
Binomial random variable
 Since p + q = 1
 For P (X < r) or P (a  X  b), the binomial sum is provided in
tables
Probability and Statistics
Binomial random variable
Example 5.5: In In a certain city district the need for
money to buy drugs is stated as the: reason for 75% of
all thefts.
Find the probability that among the next 5 theft cases
reported in this district,
a. exactly 2 resulted from the need for money to buy
drugs;
b. at most 3 resulted from the need for money to buy
drugs.
Probability and Statistics
Binomial random variable
Example 5.6: In testing a certain kind of truck tire over a
rugged terrain, it is found that 25% of the trucks fail to
complete the test run without a blowout. Of the next 15
trucks tested, find the probability that
a. from 3 to 6 have blowouts;
b. fewer than 4 have blowouts:
c. more than 5 have blowouts.
Probability and Statistics
Binomial random variable
Example 5.7: A nationwide survey of seniors by the University
of Michigan reveals that almost 70% disapprove of daily pot
smoking, according to a report in Parade.
If 12 seniors are selected at random and asked their
opinion, find the probability that the number who disapprove
of smoking pot daily is
(a) anywhere from 7 to 9:
(b) at most 5;
(c) not less than 8.
Probability and Statistics
Multinomial Distribution
If a given trial can result in k outcomes E1, E2,…,Ek with
probabilities p1, p2,,…pk, then the probability distribution of the
random variables X1, X2, …, Xk, representing the number of
occurrences for E1, E2,…,Ek in n independent trials is
with
Probability and Statistics
Multinomial Distribution
Example 5.8: According to a genetics theory, a certain cross
of guinea pigs will result in red, black, and white offspring
in the ratio 8:4:4. Find the probability that among 8
offspring 5 will be red, 2 black, and 1 white.
Probability and Statistics
Multinomial Distribution
Example 5.9: The probabilities are 0.4, 0.2, 0.3, and 0.1,
respectively, that a delegate to a certain convention
arrived by air, bus, automobile, or train. What is the
probability that among 9 delegates randomly selected at
this convention, 3 arrived by air, 3 arrived by bus, 1
arrived by automobile, and 2 arrived by train?
.
Probability and Statistics
Hypergeometric Distribution
 Similar to binomial distribution, except that it does not
require independence among trials (i.e., it can be done
without replacement)
 The probability distribution of the hypergeometric random
variable X, in which the number of successes in a random
sample of size n selected from N items of which k are labeled
success and N – k labeled failure is
Probability and Statistics
Hypergeometric Distribution
Example 5.10: To avoid detection at customs, a traveler
places 6 narcotic tablets in a bottle containing 9 vitamin
pills that are similar in appearance. If the customs
official selects 3 of the tablets at random for analysis,
what is the probability that the traveler will be arrested
for illegal possession of narcotics?
Probability and Statistics
Hypergeometric Distribution
Example 5.11: A manufacturing company uses an acceptance
scheme on production items before they are shipped. The plan is
a two-stage one. Boxes of 25 are readied for shipment and a
sample of 3 is tested for defectives. If any defectives are found,
the entire box is sent back for 100% screening. If no defectives
are found, the box is shipped.
a. What is the probability that a box containing 3 defectives will
be shipped?
b. What is the probability that a box containing only 1 defective
will be sent back for screening?
Probability and Statistics
Hypergeometric Distribution
Example 5.12: A large company has an inspection system for
the batches of small compressors purchased from vendors.
A batch typically contains 15 compressors. In the inspection
system a random sample of 5 is selected and all are tested.
Suppose there arc 2 faulty compressors in the batch of 15.
(a) What is the probability that for a given sample
there will be I faulty compressor?
(b) What is the probability that inspection will discover
both faulty compressors?
Probability and Statistics
Relationship between Hypergeometric and Binomial
Distribution
 If n is very small compared to N, hypergeometric distribution
approaches binomial distribution.
 In other words, binomial distribution is a large population
version of hypergeometric distribution.
 The quantity k/n plays the role of the binomial parameter, p
Probability and Statistics
Negative Binomial Distribution
Binomial Distribution
 Number of successes is counted for a fixed number of trials
Negative Binomial Distribution
 The trials are repeated until a fixed number of successes occur.
Probability and Statistics
Negative Binomial Distribution
 If repeated independent trials can result in a success with
probability p and a failure with probability q = 1 – p, then the
probability distribution of the random variable X, the number
of the trial on which the kth success occurs is
Probability and Statistics
Negative Binomial Distribution
Example 5.13: Suppose the probability is 0.8 that any given
person will believe a tale about the transgressions of a
famous actress. What is the probability that
a. the sixth person to hear this tale is the fourth one to
believe it?
b. the third person to hear this tale is the first one to
believe it?
Probability and Statistics
Geometric Distribution
If repeated independent trials can result in a success with
probability p and a failure with probability q = 1 – p, then the
probability distribution of the random variable X, the number of
the trial on which the first success occurs is,
Probability and Statistics
Geometric Distribution
Example 5.14: The probability that a student pilot passes
the written test for a private pilot's license is 0.7. Find the
probability that the student will pass the test
a. on the third try;
b. before the fourth try.
Probability and Statistics
Poisson Experiments
 Experiments yielding numerical values of a random variable X,
the number of outcomes occurring during a given time
interval or in a specified region
 A specified region could be a line segment, an area, a volume,
or perhaps a piece of material
Probability and Statistics
Properties of Poisson Process
1. The number of outcomes occurring in one time interval or
specified region is independent of the number that occurs in
any other disjoint time interval or region of space. In other
words, Poisson process has no memory.
2. The probability that a single outcome will occur during a very
short time interval or in a small region is proportional to the
length of the time interval or the size of the region and does
not depend on the number of outcomes occurring outside
this time interval or region.
3. The probability that more than one outcome will occur in
such a short time interval or fall in such a small region is
negligible.
Probability and Statistics
Poisson Distribution
 The probability distribution of the Poisson random variable X,
representing the number of outcomes occurring in a given
time interval or specified region denoted by t, is
  - the average number of outcomes per unit time or region
Probability and Statistics
Poisson Distribution
Example 5.15: On average a certain intersection results in 3
traffic accidents per month. What is the probability that
for any given month at this intersection
a. exactly 5 accidents will occur?
b. less than 3 accidents will occur?
c. at least 2 accidents will occur?
Probability and Statistics
Poisson Distribution
Example 5.16: A secretary makes 2 errors per page, on
average. What is the probability that on the next page he
or she will make
(a) 4 or more errors?
(b) no errors?
Probability and Statistics
Poisson Distribution and binomial distribution
As n   and p  0, and np remains constant, Binomial
Distribution approaches a Poisson Distribution where np = t
Probability and Statistics
Poisson Distribution and binomial distribution
Example 5.17: In a manufacturing process where glass products
are produced, defects or bubbles occur, occasionally rendering
the piece undesirable for marketing. It is known that, on average,
1 in every 1000 of these items produced has one or more
bubbles. What is the probability that a random sample of 8000
will yield fewer than 7 items possessing bubbles?
Some Continuous Probability
Distributions