L4 - The Binomial and the Poisson Distributions Flashcards Preview

18ECA005 - Data Analysis II > L4 - The Binomial and the Poisson Distributions > Flashcards

Flashcards in L4 - The Binomial and the Poisson Distributions Deck (10)
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What are Factorials?

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)

0! = 1


What is a combination?

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


What is the combination formula?

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


What is the Binomial Distribution?

- 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.


What is the Probability Function of Binomial Distribution?

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

- 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)


How can we write the Binomial Distribution?

- 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.


What is the Poisson Distribution?

- 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.


What is the Poisson Distribution given by?

- 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


What was the most common use of the Poisson Distribution?

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
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.


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

- 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.