Chapter 3: Discrete Random Variable - Binomial Probability Distribution - Hype Hyperg rgeome eometry try Dist Distributi ribution on - Poisson Distribution
3.1 Definition A random variable is a variable whose value is a numerical numer ical outcome of a random phenomenon. As a real-v real-valued alued functi function, on, random random variabl variable e often often describes describes some 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.
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
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:
0 1 2) = = 3) () = 1 1)
all
and
Example 3.1
1) Find the value of k for a given probability distribution function.
= for = 0,1,2,3,4
Ans: Since
= 1
= 20.
Therefore
all
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
≠ 1
all
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
1 1 1 1 0 = = 0 = 2 × 2 × 2 = 8 1 1 1 3 1 = = 1 = 2 × 2 × 2 × 3 = 8 1 1 1 3 2 = = 2 = 2 × 2 × 2 × 3 = 8 1 1 1 1 3 = = 3 = 2 × 2 × 2 = 8
Thus, the probability distribution of X is
()
0
1
2
3
1/8
3/8
3/8
1/8
3.3 Cumulative Distribution Function(CDF) of Discrete Random Variable
Cumulative distribution function(CDF)of a discrete random variable 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
()
0
1
2
3
4
0.41
0.37
0.16
0.05
0.01
Construct the cumulative distribution function of X.
Ans:
(), find =2 >1 ( 3) ( < 2) (0 < < 3) (2 < 4) (1 3)
Then, using a) b) c) d) e) f) g)
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
() = () ∙ all
g
is
3.5 Variance of Discrete Random Variable
= = 2 = 2 = 2 = where = , = = Variance ,
Standard deviation,
=
Example 3.3: The random variable X, representing the number of errors per 100 lines of software code, has the following probability distribution:
()
2
3
4
5
6
0.01
0.25
0.4
0.3
0.04
Find the mean and variance of X . Ans:
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 classified as a success or a failure. The probability of success, denoted by p, remains constant from trial to trial. The repeated trials are independent.
Binomial Probability Distribution:
~(,)
Notation: The probability of obtaining by
successes from trials is given = −
where = total number of trials = probability of success ; probability of failure num. of successes in trials
=1 =
For Binomial Distribution:
= Variance, =
Mean,
Example 3.4: 1) The probability that a patient recovers from a delicate heart 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 that a) 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)
3.7 Hypergeometric Distribution •
A discrete probability distribution that describes the probability 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.
Hypergeometric Distribution A sample of size n is selected from N items of which k are labelled success and N-k labelled failure. The probability of the number of success obtained from the random sample of size n is
= ,
= 0,1,2,…,
For Hypergeometric Distribution:
= Variance, = 1 where = Mean,
− −
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)
Example 3.5: 3) A distributor buys 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? 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)
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 1 person with blood type O, 2 with blood type A, and 2 with blood type B? (Ans: 3/14)
3.8 Poisson Distribution •
A discrete probability distribution that expresses the probability 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.
Poisson Distribution The probability of a given number of outcomes occurring 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:
= Variance, =
Mean,
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;)
Example 3.6:
3) In the inspection of paper produced by a machine, 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)