discrete random_variables

CHAPTER 3
DISCRETE RANDOM VARIABLES
Prepared by: Noryani Muhammad

EACH NEW DAY IS ANOTHER CHANCE TO CHANGE YOUR LIFE
OBJECTIVES
1. To study the concept of discrete random variables.
2. To construct the probability and cumulative
distribution functions, mean, and standard deviation.
3. To study and apply the binomial, hypergeometric,
and poisson distributions.

3.1 CONSEPT OF A RANDOM VARIABLE
• A quantitative variable whose value is random,
determined by the outcome of a random
experiment.
• Denoted by uppercase X and the possible values
are denoted by lowercase x.
• Discrete random variable assume only
countable value.

3.2 PROBABILITY DISTRIBUTION FUNCTION
1)(
1)(0


xXp
xXp
allx
Two conditions of a probability distribution function:
Example:

3.2 PROBABILITY DISTRIBUTION FUNCTION

Test your understanding….

Determine whether or not each table represents a valid probability distribution.

3.3 CUMULATIVE DISTRIBUTION FUNCTION
The cumulative distribution function f(x) of a discrete random
variable X with the probability distribution function f(x) is
Example ~ Refer example 3.3

3.3 CUMULATIVE DISTRIBUTION FUNCTION
Example 3.5

Excellent does not an accident, but it comes through a hard work!!
1. Find the cumulative distribution function of X:
Breakdowns per week 0 1 2 3
p.d.f , p(X=x) 0.15 0.2 0.35 0.3
c.d.f, p(X=x)
2. The probability distribution of X, the number of imperfections per 10
meters of a synthetic fabric in continuous rolls of uniform width, is given by:
a) Construct the cumulative distribution function of X.
b) Using the answer in (a), find the probability that there is at least three
imperfections per 10 meters of the fabric.

3.4 EXPECTED VALUE AND VARIANCE
Expected value: Value that is expected to occur
every time when experiment is repeated.
Variance: Deviation from the mean.

3.4 EXPECTED VALUE AND VARIANCE
Example: Find the mean and standard deviation of X

Find the mean and standard deviation of X:

3.5 THE BINOMIAL DISTRIBUTION
The binomial distribution is applied to find the
probability that an outcomes will occurs x times in n
performances of an experiment which satisfies the
Bernoulli Process.
1. The experiment consists of n repeated trials.
2. Each trial results in two outcomes only: classified as a success (p) or
a failure (q).
Success = the outcome to which the question refers
Failure=the outcome to which it does not refer
3. The probability p and q remains constant from trial to trial.
4. The repeated trials are independent.

Examples
1. Testing the effectiveness of a drug: Several patients take the drug
(the trials), and for each patient the drug is either effective or not
effective (two possible outcomes).
2. Weekly sales of a car salesperson: The salesperson has several
customers during the week (the trials), and for each customer the
salesperson either makes a sales or does not make a sale (two possible
outcomes).
3. Taste tests for colas: A number of people taste two different colas
(the trials), and for each person the preference is either for the first
cola or for the second cola (two possible outcomes).

For a binomial experiment, the probability of exactly
x successes in n trials is given by the binomial formula:

Examples
1. The probability that a certain kind of component will survive a given
shock test is 3/4. If 10 components are tested,
(a) determine the probability distribution function for a component tested
that survived.
(b) find the probability that exactly 5 components tested that survived.
(0.0584)
2. The probability that an engineering student will get an ‘A’ for a final year
project is 0.75. Find the probability that
a. exactly 2 of the next 4 students will get an ‘A’ for the project. (0.2109)
b. at most 1 of the next 4 students will get an ‘A’ for the project. (0.0508)
c. at least 3 of the next 4 students will get an ‘A’ for the project. (0.7383)

Exercise 3.7 (pg65)
A large chain retailer purchases a certain kind of electronic device from a
manufacturer. The manufacturer indicates that the defective rate of the device is 3%.
The inspector of the retailer randomly picks 20 items from a shipment.
a. What is the probability that there will be at least one defective item? (0.4562)
b. How many defective items are expected from this shipment? (0.6)
Exercise 3.8 (pg66)
It is conjectured that an impurity exists in 30% of all drinking wells in a certain rural
community. In order to gain some insight on this problem, it is determined that some
tests should be made. It is too expensive to test all of the many wells in the area so 10
were randomly selected for testing.
(a) What is the probability that exactly three wells have the impurity assuming that
the conjecture is correct? (0.2668)
(b) What is the probability that more than three wells are impure? (0.3504)
(c) How many wells are expected to have the impurity? (3)
(d) Find the standard deviation of the wells which have the impurity. (1.4491)

3.6 THE HYPERGEOMETRIC DISTRIBUTION
The probability of x successes in n draws without replacement
from a finite population of size N containing exactly k successes.
nx
n
N
xn
kN
x
k
xp ,...,2,1,0)( 





















   















N
k
pwhere
N
nN
ppnVariance
npMean
,
1
1:
:
2



3.6 THE HYPERGEOMETRIC DISTRIBUTION
Example
1. A shipment of 20 computer chips contains 5 that are defective. If 10 of
them are randomly chosen for inspection, what is the probability that
2 of them will be defective? (0.3483)
2. Batches of 40 digital voice recorders each are considered acceptable
if they contain no more than 3 defectives. The procedure for sampling
the batch is to select 5 recorders at random without replacement and
to reject the batch if a defective is found.
a. What is the probability that exactly 1 defective is found in the sample
if there are 3 defectives in the entire batch? (0.3011)
b. Find the mean and variance of the random variable above. (0.375,
0.3113)

Exercise 3.9 (pg69)
A distributor sells 100 machine components from a local manufacturer and 200
machine components from a foreign manufacturer. If four components are selected
randomly and without replacement,
(a) what is the probability that they are all from the local manufacturer? (0.0119)
(b) what is the probability that two or more components in the sample are
from the local manufacturer? (0.4075)
(c) what is the probability that at least one component in the sample is
from the local manufacturer? (0.8045)
Exercise 3.10 (pg69)
Five new recruits from different family and education background have been trained
separately to conduct a special task. After they have completed their training, they
were sent to join other recruits. A random sample of 10 of these recruits is selected.
Let X = the number of new recruits in the second sample. If there are actually 25
recruits altogether, what is the probability that:
a. the number of new recruits in the second sample is two? (0.3854)
b. the number of new recruits in the second sample is at most two? (0.6988)

3.7 THE POISSON DISTRIBUTION
Poisson experiments are experiments yielding numerical values
of a random variable X, the number of outcomes occurring
during a given unit of time or space.
3 conditions to apply Poisson Probability Distribution:
a. X is a discrete random variable.
b. The occurrences are random.
c. The occurrences are independent.
Example :
 The number of UTeM students completing their industrial
training in four months period.
 The number of new PROTON cars sold yearly.
 The parts per million of some toxin found in the water or air
emission from a manufacturing plant.

!
)(
x
e
xp
x 
 

Where:
λ=no of occurrence in the interval
e=2.71828






:
:
:
2
ionStndDeviat
Variance
Mean

3.7 THE POISSON DISTRIBUTION
Examples
1.
Over a 10-minute period, a counter records an average of 1.3 gamma
particles per millisecond coming from a radioactive substance. To a good
approximation, the distribution of the count, X, of gamma particles
during the next millisecond is Poisson. Determine:
(a) the probability of one or more gamma particles. (0.7275)
(b) the variance. (1.3)
2. If a bank receives on the average λ = 6 bad checks per day, what are the
probabilities that it will receive:
(a) 4 bad checks on any given day? (0.1339)
(b) 10 bad checks over any 2 consecutive days? (0.1048)

In the inspection of tin plate produced by a continuous electrolytic
process, 0.2 imperfection is spotted per minute, on average. Find
the probabilities of spotting
(a) one imperfection in 3 minutes. (0.3293)
(b) at least two imperfections in 5 minutes. (0.2642)
(c) at most one imperfection in 15 minutes. (0.1992)
Three is the average number of oil tankers arriving each day at a
small port. The facilities at the port can handle at most 5 tankers per
day. What is the probability that on a given day tankers have to be
turned away? (0.084)

Excellent does not an accident, but it comes through a hard work!!28

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