View
268
Download
0
Category
Preview:
Citation preview
7/26/2019 Chapter 3 Discrete Random Variable.pdf
http://slidepdf.com/reader/full/chapter-3-discrete-random-variablepdf 1/23
Chapter 3: Discrete Random
Variable
- Binomial Probability Distribution
- Hypergeometry Distribution
- Poisson Distribution
7/26/2019 Chapter 3 Discrete Random Variable.pdf
http://slidepdf.com/reader/full/chapter-3-discrete-random-variablepdf 2/23
3.1 Definition
A random variable is a variable whose value is a numerical
outcome of a random phenomenon.
As a real-valued function, random variable often describessome numerical quantity of a given event. For example, the
number of heads after a certain number of coin flips.
X variable
x possible value of the random variable
A random variable is called a discrete random variable if its
set of possible outcomes is countable.
7/26/2019 Chapter 3 Discrete Random Variable.pdf
http://slidepdf.com/reader/full/chapter-3-discrete-random-variablepdf 3/23
Example:
Two balls are drawn in succession without replacement from
an urn containing 4 red balls and 3 black balls. Let Y denotes
the number of red balls, the values y are
Sample Space y
RR 2
RB 1
BR 1
BB 0
7/26/2019 Chapter 3 Discrete Random Variable.pdf
http://slidepdf.com/reader/full/chapter-3-discrete-random-variablepdf 4/23
3.2 Probability Mass Function (pmf)
• A function that gives the probability that a discrete
random variable is exactly equal to some value.
• Also known as probability function or probability
distribution of the discrete random variable X
• Properties:
1) 0 12)
= = and
3) all
() = 1
7/26/2019 Chapter 3 Discrete Random Variable.pdf
http://slidepdf.com/reader/full/chapter-3-discrete-random-variablepdf 5/23
Example 3.1
1) Find the value of k for a given probability distribution
function.
= for = 0,1,2,3,4Ans: Since
all
= 1
Therefore = 20.
2) Check whether the following can be defined as a
probability mass function. Explain your answer.
= + for = 1,2,3,4,5
Ans: No, since
all ≠ 1
7/26/2019 Chapter 3 Discrete Random Variable.pdf
http://slidepdf.com/reader/full/chapter-3-discrete-random-variablepdf 6/23
3) A fair coin is tossed three times. Find the probability
distribution for the number of heads obtained.
Ans: Let X be the number of heads obtained
0 = = 0 = 12 × 1
2 × 12 =
18
1 = = 1 = 12 × 12 × 12 × 3 = 38 2 = = 2 = 1
2 × 12 × 1
2 × 3 = 38
3 = = 3 =
1
2 ×
1
2 ×
1
2 =
1
8Thus, the probability distribution of X is
0 1 2 3
() 1/8 3/8 3/8 1/8
7/26/2019 Chapter 3 Discrete Random Variable.pdf
http://slidepdf.com/reader/full/chapter-3-discrete-random-variablepdf 7/23
3.3 Cumulative Distribution Function(CDF) of Discrete Random Variable
Cumulative distribution function(CDF)- of a discrete randomvariable with probability distribution function is
= = ≤
, ∞ < < ∞Example 3.2:
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
Construct the cumulative distribution function of X.
0 1 2 3 4
()0.41 0.37 0.16 0.05 0.01
7/26/2019 Chapter 3 Discrete Random Variable.pdf
http://slidepdf.com/reader/full/chapter-3-discrete-random-variablepdf 8/23
Ans:
Then, using (), finda) = 2b) > 1c)
( 3)d) ( < 2)e) (0 < < 3)f) (2 < 4)g) (1 3)
7/26/2019 Chapter 3 Discrete Random Variable.pdf
http://slidepdf.com/reader/full/chapter-3-discrete-random-variablepdf 9/23
3.4 Expected Value of Discrete Random Variable
Definition:
For a discrete random variable X with probability distribution
(),
• the mean, or expected value of random variable X is
= = all
∙
•the mean, or expected value of random variable g is
() = all
() ∙
7/26/2019 Chapter 3 Discrete Random Variable.pdf
http://slidepdf.com/reader/full/chapter-3-discrete-random-variablepdf 10/23
3.5 Variance of Discrete Random Variable
Variance, = = 2
=
2
= 2 =
where = , = =
Standard deviation, =
7/26/2019 Chapter 3 Discrete Random Variable.pdf
http://slidepdf.com/reader/full/chapter-3-discrete-random-variablepdf 11/23
Example 3.3:
The random variable X, representing the number of
errors per 100 lines of software code, has the following
probability distribution:
Find the mean and variance of X .
Ans:
2 3 4 5 6
() 0.01 0.25 0.4 0.3 0.04
7/26/2019 Chapter 3 Discrete Random Variable.pdf
http://slidepdf.com/reader/full/chapter-3-discrete-random-variablepdf 12/23
3.6 Binomial Distribution
The Binomial process possess all the following properties:
• The experiment consists of n repeated trials
•
Each trials results in an outcome that may be classifiedas a success or a failure.
• The probability of success, denoted by p, remains
constant from trial to trial.
• The repeated trials are independent.
7/26/2019 Chapter 3 Discrete Random Variable.pdf
http://slidepdf.com/reader/full/chapter-3-discrete-random-variablepdf 13/23
Binomial Probability Distribution:
Notation: ~(, )The probability of obtaining successes from trials is given
by
= −
where = total number of trials
= probability of success
= 1 ; probability of failure
=num. of successes in
trials
For Binomial Distribution:
Mean, = Variance, =
7/26/2019 Chapter 3 Discrete Random Variable.pdf
http://slidepdf.com/reader/full/chapter-3-discrete-random-variablepdf 14/23
Example 3.4:
1) The probability that a patient recovers from a delicateheart operation is 0.9. What is the probability that
exactly 5 of the next 7 patients having this operation
survive? Find the number of surviving patients that is
expected from this sample. (Ans: 0.1240)
2) It is known that 60% of mice inoculated with a serum
are protected from a certain disease. If 5 mice are
inoculated, find the probability thata) None contracts the disease; (Ans: 0.0778)
b) Fewer than 2 contract the disease; (Ans: 0.3370)
c) More than 3 contract the disease. (Ans: 0.0870)
7/26/2019 Chapter 3 Discrete Random Variable.pdf
http://slidepdf.com/reader/full/chapter-3-discrete-random-variablepdf 15/23
3.7 Hypergeometric Distribution
• A discrete probability distribution that describes theprobability of x successes in n draws, without
replacement, from a finite population of size N
containing exactly k successes.
The Hypergeometric process possess all the following
properties:
• A random sample of size n is selected without
replacement from N items.• Of the N items, k may be classified as success and N-k
are classified as failure.
7/26/2019 Chapter 3 Discrete Random Variable.pdf
http://slidepdf.com/reader/full/chapter-3-discrete-random-variablepdf 16/23
Hypergeometric Distribution
A sample of size n is selected from N items of which k are
labelled success and N-k labelled failure. The probabilityof the number of success obtained from the random
sample of size n is
=
, = 0,1,2, … ,
For Hypergeometric Distribution:
Mean, = Variance, = 1 −
−where
=
7/26/2019 Chapter 3 Discrete Random Variable.pdf
http://slidepdf.com/reader/full/chapter-3-discrete-random-variablepdf 17/23
Example 3.5:
1) A homeowner plants 6 bulbs selected randomly from a
box containing 5 tulip bulbs and 4 daffodil bulbs. What is
the probability that he planted 2 daffodil bulbs and 4 tulip
bulbs? (Ans: 5/14)
2) If 6 of 18 new buildings in a city violate the building code,what is the probability that a building inspector, who
randomly selects 4 of the new buildings for inspection,
will catch
a) None of the buildings that violate the building code?b) 2 of the new building violate the building code?
c) At least 3 of the new buildings that violate the
building code?
(Ans: 0.1618; 0.3235; 0.0833)
7/26/2019 Chapter 3 Discrete Random Variable.pdf
http://slidepdf.com/reader/full/chapter-3-discrete-random-variablepdf 18/23
Example 3.5:
3) A distributor buys 100 machine components from a localmanufacturer 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 localmanufacturer?
b) What is the probability that two or more components
in the sample are from the local manufacturer?
c) What is the probability that at least one component in
the sample is from the foreign manufacturer?
(Ans: 0.0119; 0.4075; 0.9881)
7/26/2019 Chapter 3 Discrete Random Variable.pdf
http://slidepdf.com/reader/full/chapter-3-discrete-random-variablepdf 19/23
Example 3.5:
Multivariate Hypergeometric Distribution:4) A group of individuals is used for a biological case study.
The group contains 3 people with blood type O, 4 with
blood type A, and 3 with blood type B. What is the
probability that a random sample of 5 will contain 1person with blood type O, 2 with blood type A, and 2
with blood type B?
(Ans: 3/14)
7/26/2019 Chapter 3 Discrete Random Variable.pdf
http://slidepdf.com/reader/full/chapter-3-discrete-random-variablepdf 20/23
3.8 Poisson Distribution
• A discrete probability distribution that expresses theprobability of a given number of events occurring in a
fixed interval of time and/or space if these events
occur with a known average rate and independently of
the time.
The Poisson process possess all the following properties:
• Occurrence in a given time interval is independent to
occurrence in other time intervals.• Probability of more than one success in given time
interval is negligible.
7/26/2019 Chapter 3 Discrete Random Variable.pdf
http://slidepdf.com/reader/full/chapter-3-discrete-random-variablepdf 21/23
Poisson Distribution
The probability of a given number of outcomesoccurring in a given time interval or specified region
is given by
= −
! , = 0,1,2, … ,where is the average number of outcomes per unit
time, distance, area or volume.
For Poisson Distribution:
Mean, = Variance, =
7/26/2019 Chapter 3 Discrete Random Variable.pdf
http://slidepdf.com/reader/full/chapter-3-discrete-random-variablepdf 22/23
Example 3.6:
1) If a bank receives on the average 6 bad cheques per
day, what are the probability that it will receive
a) 4 bad cheques on any given day?
b) 10 bad cheques over any 2 consecutive days?
(Ans: 0.1338; 0.1048)
2) At a checkout counter customers arrive at an
average of 1.5 per minute. Find the probability that
a) At most 4 will arrive in any given minute;
b) At least 3 will arrive during an interval of 2
minutes;
(Ans: 0.9814; 0.5768;)
7/26/2019 Chapter 3 Discrete Random Variable.pdf
http://slidepdf.com/reader/full/chapter-3-discrete-random-variablepdf 23/23
Example 3.6:
3) In the inspection of paper produced by amachine, 0.2 imperfection is spotted per minute
on average. Find te probabilities of spotting
a) One imperfection in 3 minutes;
b) At least two imperfections in 5 minutes;
c) At most one imperfection in 15 minutes.
(Ans: 0.3293; 0.2642; 0.1991)
Recommended