L4 - The Binomial and the Poisson Distributions

Question 1

What are Factorials?

Answer 1

Factorials are, very simply, products. They are written with an ! sign and:
1! = 1 --> (1 factorial)
2! = 2*1 = 2 --> (2 factorial)
3! = 3*2*1= 6 --> (3 factorial)
4! = 4*3*2*1=24 -->  (4 factorial)

Note:
0! = 1

Question 2

What is a combination?

Answer 2

A combination is the number of ways that x objects can be selected
from a set of n. No object can be chosen more than once but we are not
concerned about the order.
-Ex: how many ways are there to have 3 girls out of 5 children?
- List all outcomes and count : GGGGG, GGGGB… but there are 2^5 of them

Question 3

What is the combination formula?

Answer 3

nCx = (n!)/[x!(n-x)!]

where n is the sample size and x is the number of outcomes

• this only gives you the number of times the event will occur
• to get the probability you simple give this number be the total outcome

Question 4

What is the Binomial Distribution?

Answer 4

• Take a random experiment with only 2, mutually exclusive outcomes, called success and failure, with probabilities P and 1-P
  (eg child is either a girl or a boy).
• If we repeat that experiment n times (i.e. we have n trials) and P
  remains constant (the outcomes are independent) the resulting distribution of successes is a binomial distribution.
• So the binomial models the probability of x ‘successes’ from n
  independent trials
• This helps in problems like “what is the probability of having 3 girls out of 5 children” where we would have x = 3 and n = 5.

Question 5

What is the Probability Function of Binomial Distribution?

Answer 5

P(x)=(nCx) x (P^x) x [(1-P)^nx]

Where:

• n is the number of trials (in the example, 5);
• x is the number of successes (in the example number of girls);
• P is the probability of a success in a single trial (in the example, 0.5)

Question 6

How can we write the Binomial Distribution?

Answer 6

• n and P are the parameters that define the binomial distribution, so we
  can write –> X~B(n,P)
• The mean and variance of a binomially distributed variable can be showed to be
• μ = nP
• σ^2 = nP(1-P)
• The probabilities of a binomial distribution can also be found on statistical tables.

Question 7

What is the Poisson Distribution?

Answer 7

• Waiting or queuing problems: x is the number of occurrences in a
  certain interval of time. These are known as count data. The occurrences
  must always be non negative integers (0,1,2,3 etc) and they are
  proportional to the length of time considered.
• As a limiting case of the binomial, when P is very small and n is very
  large. It can be much easier in this case to use the Poisson as the calculations are faster.

Question 8

What is the Poisson Distribution given by?

Answer 8

• P(x)= [(μ^x) x (e^-μ)]/(x!)

E(x)=μ and var(x)=μ

The mean µ is the only parameter that characterizes the Poisson
distribution

Question 9

What was the most common use of the Poisson Distribution?

Answer 9

The most common use of the Poisson distribution is for events that
occur over time:
- The number of buses that arrive at a certain stop in an hour;
- The number of phone calls at a switchboard at a certain time in a day;
- The number of times a piece of equipment fails during a 6 months
period;
-
We know what the average number of occurrences is over that period of time (µ), and this number is proportional to the length of time considered: for instance if on average there are 4 buses per hour, then on average there is one every 15 minutes.

Question 10

When is the Poisson Distribution a goof Approximation to the Binomial Distribution?

Answer 10

• If x is a binomial we know that E(x) = nP.
• If n is very large and P very small, in particular if nP ≤ 7 then the Poisson is a good approximation to the binomial.
• In this case x follows a Poisson with mean µ= nP.