Lecture 7- Discrete Probability Distributions

Question 1

A random variable

Answer 1

one whose value is determined by the outcome of a random experiment.

Question 2

A discrete random variable

Answer 2

assumes countable variables

Question 3

A continuous random variable

Answer 3

assumes values that can be measured or a value that is contained in one or more intervals.

Question 4

A probability distribution

Answer 4

links a random variable X with the probability that X assumes a discrete value or a range of values. This can be presented by a table, function or formula
*Random variables can be discrete or continuous
*Probability distributions are also correspondingly discrete or continuous

Question 5

Discrete vs Continuous Random Variables

Answer 5

Every random variable has associated with it a probability distribution.
*discrete random variables possess discrete probability distributions
*continuous random variables possess continuous probability distributions.

Question 6

Discrete Probability Distributions covered in this course

Answer 6

The Binomial Distribution
*The Poisson Distribution
the Approximation of the Binomial Distribution by the Poisson Distribution.

Question 7

Continuous Probability Distributions covered in this course

Answer 7

The Normal Distribution
*The Student-t Distribution
*The Chi Square Distribution
*The F Distribution

Question 8

Two types of notation

Answer 8

Factorials
–The Combination Notation

Question 9

Factorials

Answer 9

The value of the factorial of a number is obtained by multiplying all the integers from that number to 1
*The symbol n!, read as “n factorial”, represents the product of all the integers from n to 1.
*In other words,n! = (n) (n-1) (n-2)(n-3)…….3.2.1
*By definition, 0! = 1
*For example
–6! = 6x5x4x3x2x1
–10! = 10x9x8x7x6x5x4x3x2x1

Question 10

Combinations

Answer 10

Combinations give the number of ways r elements can be selected from n elements.
*The number of Combinations for selecting r from n distinct elements is given by:
*nCr = n !
r! ( n-r)!
*5C2 is therefore calculated as (5!) / (2! 3!)
= 5 x 4 x 3 x 2 x 1
(2x1) (3x2x1)
= 120/12 = 10
All Scientific Calculators possess the “Combination” Formula

Question 11

Binomial Experiment

Answer 11

one that possesses the following properties:
–The experiment consists of n repeated trials, n = 0, 1, 2, …
–Each trial results in an outcome that may be classified as a success or a failure
–The probability of a success, denoted by p, remains constant from trial to trial (in our illustration, p = 1/2)
–The repeated trials are independent
If any ONE of these conditions is violated, we do NOT have a Binomial Experiment.

Question 12

Example of binomial experiment (tossing a fair coin)

Answer 12

P(Heads) = ½, P(Tails) = ½
*If you were to toss the coin 10 times, would you get Heads for 5 of those times?
*The truth is that you can get Heads any amount of times ranging from 0 to 10
*Of course, you are more likely to get 5 than 0, but the fact remains that both these numbers (as well as others in the range 0-10) are possible
*So the question is, how probable is each one?
*In other words, if we define the random variable X as the number of Heads we get, can we create a probability distribution for X?
the experiment is the tossing of the coin
*You are doing 10 trials of this experiment (tossing the coin 10 times)
*Each time you toss the coin (for each trial of the experiment), only one of two outcomes is possible: Heads (H) or Tails (T)
*No matter the previous outcomes, for each time you toss the coin, the P(H) remains ½ and the P(T) remains ½

Question 13

Example of binomial experiment pt 2

Answer 13

Tossing a Fair Coin 30 times, defining X as the r.v. the number of Heads, and we are interested in creating the probability distribution for X
–The experiment (tossing the coin) consists of n repeated trials, n = 0, 1, 2, … (n=10)
–Each trial (tossing the coin) results in an outcome that may be classified as a success (Heads) or a failure (Tails)
–The probability of a success, denoted by p, remains constant from trial to trial (the probability of getting Heads remains constant each time, and p= ½)
–The repeated trials are independent (however many Heads we get on previous trials, that is not affecting the probability of getting Heads in future trials

Question 14

The Binomial Distribution Formula

Answer 14

The formula is developed out of our attempt to put meaning to the question
‘What does P(X = r) mean?’
*P(X = r) ≡ P(r successes in n trials)
≡ P(r successes and n-r failures)
*The event of ‘r successes and n-r failures’ can occur in several ways; each way is called a combination of r successes and n-r failures.
*In our example, suppose we wanted to know P(X=2) : in other words, when tossing the coin 10 times, what is the probability that we get 2 Heads?
*If we get 2 Heads (r successes), that means we are getting 8 Tails (n-r failures)

Question 15

The Binomial Distribution Formula

Answer 15

The formula is developed out of our attempt to put meaning to the question
‘What does P(X = r) mean?’
*P(X = r) ≡ P(r successes in n trials)
≡ P(r successes and n-r failures)
*The event of ‘r successes and n-r failures’ can occur in several ways; each way is called a combination of r successes and n-r failures.
*In our example, suppose we wanted to know P(X=2) : in other words, when tossing the coin 10 times, what is the probability that we get 2 Heads?
*If we get 2 Heads (r successes), that means we are getting 8 Tails (n-r failures
If X is a random variable that follows a Binomial Distribution, then
P(X = r) = nCr (p)r (q)n-r
*n = no of trials of the experiment
*p = probability of a success
*q = probability of a failure
*NOTE: Recognise the “Combination” Formula we previously introduced?

Question 16

The parameters of the binomial distribution

Answer 16

The Parameters of the Binomial Distribution
*P(X = r) = nCr (p)r (q)n-r
*n and p are called the parameters of the Binomial Distribution
*Without values for n and p we cannot use the Binomial Distribution formula to compute P(X = r).
*We therefore say that if the random variable X follows a Binomial Distribution:
X Bin (n,p)
P(X = r) = nCr (p)r (q)n-r

Question 17

Moments of the Binomial Value

Answer 17

Like any other random variable, we are interested in the Expectation and Variance of the Binomial Variable
*These are also called the Moments of First and Second Order
Expectation μ = E(X) = np
Variance σ2 = Var(X) = npq Standard Deviation σ = ( npq)

Question 18

The Poisson Distribution

Answer 18

This is another important probability distribution of a discrete random variable that has a large number of applications
*It is applied to experiments with random and independent occurrences. The occurrences are always considered with respect to an interval
*The interval may be a time interval, a space interval, or a volume interval

Question 19

Conditions to Apply the Poisson Distribution

Answer 19

The following three conditions must be satisfied to apply the Poisson probability distribution:
–X is a discrete random variable
–A particular occurrence is recorded in a particular interval
–The occurrences are random
–The occurrences are independent
The Poisson Process fits events that are randomly scattered over time and/or space (i.e. you cannot predict when or where an event will occur).

Question 20

Examples of the Poisson Distribution

Answer 20

The number of phone calls received by a household during a given day
–The receiving of the call is the occurrence, the interval is one day, the occurrences are random
*The number of accidents that occur on a given highway during a one week period
–An accident is an occurrence, the interval is 1 week, the occurrences are random
*The number of defective items in the next 100 items manufactured on a machine
–A defective item is an occurrence, the interval is every 100 items, the occurrences are random
In Contrast….
*The arrival of patients at a doctor’s office: these are non-random if the patients have to make appointments to see the doctor
*The arrival of commercial planes at an airport: these are non-random since planes are scheduled to arrive at certain times

Question 21

The Poisson Distribution Formula

Answer 21

In the terminology of the Poisson Distribution, the average number of occurrences in a given interval is denoted by (greek letter lambda)
*The actual number of occurrences in that interval is denoted by r
*Then, using the Poisson probability distribution, we find the probability of r occurrences during an interval, given that the mean number of occurrences during that interval is (weird sign)
If the random variable X follows a Poisson Distribution, this means that X (λ)
*The probability of r occurrences in an interval is:
P( X = r) = e – λ λ r r = 0, 1, 2,……
r!
where e is approximately the value 2.718 (look for it on your calculator) and λ is the mean number of occurrences in that interval
*By this formula, once we know the value of λ we can compute any probability under the Poisson Distribution
*λ is therefore the “parameter” of the Distribution
The Poisson Distribution Formula

Question 22

Moments of the Poisson Distribution Formula

Answer 22

If the random variable X follows a Poisson Distribution:
X (λ)
P( X = r) = e – λ λ r r = 0, 1, 2,……
r!
*What are the moments of first and second order?
*If X Pr(λ), then:
E(X) = λ
Var(X) = λ
SD(X) = λ

Question 23

The Approximation Of The Binomial Distribution By The Poisson Distribution.

Answer 23

When q is approximately equal to 1, we can be reasonably accurate in using the Poisson Distribution as an approximation to the Binomial Distribution.
*
*The Poisson Distribution yields a good approximation to the Binomial Distribution when n is large and p is relatively small (i.e. p close to zero).
*To execute the approximation we simply set the parameter λ = np and substitute into the Poisson Distribution Formula.

Question 24

Comparative Probabilities of P(X = 50) when p = 0.02 for three values of n viz. 50, 100 and 200

Answer 24

n Binomial Poisson Error % Error
50 0.0027 0.0031 .0004 15
100 0.0353 0.0361 .0008 2
200 0.1579 0.1563 .0016 1
Clearly, as n gets larger, the two yield approximately the same values
*This approximation of the Poisson to the Binomial was of great practical importance before the advent of the electronic computer.
*Now it is less important but it is still intellectually interesting
*We will soon consider the much more appealing approximation of the Binomial distribution by the Normal Distribution