Chapter 2) Discrete Distrubutions : Binomial And Poisson

Question 1

Conditions for binomial probability
(3)

Answer 1

1) n independent trials
2) only two possibilities , success and failure
3) probability of these are FIXED

Question 2

What is the mean and the variance of a binomial distribution?

Answer 2

Mean or exoected value = NP

Variance = NPQ (n x p x (1-p))

Question 3

How to find the most probable value in binomial and why is this only one

Answer 3

Go around the mean, and test for the highest probability

Will only be one because binomial distribution is SYMMETRICAL and has one peak

Question 4

In order to derrive E(x) and var(x) for binomial we need to know what binomial distribution basically is

What is a Bernoulli and how does the binomial distribution x relate to this

Answer 4

It’s the sum of n Independent Bernoulli trials

Where a Bernoulli trial is a special case of thr binomial distribution where n =1 and p = p

Assuming this we can say that The whole distributor of X is a summation of all n Bernoulli trials
X = x1 +x2 +x3 to n

Question 5

Derrive e(x) and var(x) taking into account Bernoulli

Answer 5

If x -B (1,p) there are only two possibilites X = 0 and x=1

Write these down in table form and write the probabilities for both of them. ( Q, P)

Find E (xi)

Relate to equation E(x) = e(x1 +x2 +x3) = E(x1) + E(x2) …

Realise this means multiply by n

2) do the same for Var( x), and factorise

Will get np and np (1-p)

Question 6

what does poisson distrubtuin normally model

Answer 6

Things with uniform average rate of occurrence
And that are modelled to infinity (not fixed number of trials)

Question 7

What are THREE conditions that possion distribution can be used
IMPORTANT

Remember to use in context!!

Answer 7

1) events are RANDOM and INDEPENDENT
2) events occur SINGLY (2,3 can’t happen at once)
3) AND OCCUR AT A UNIFORM AVERAGE rate of occurrence = to LAMDA

Question 8

So why is insurance claims after a flood not poission

Answer 8

Because these aren’t Gonna happen at a uniform rate of occurrence, they have been spiked up because of the flood!

Normally then yeah can use

Question 9

What else about E(X) and Var(x) , what are they, and how can you thus tell based on these model could fit position distribution

Answer 9

E(x) = lambda
Var(x) ALSO = LAMBDA

so if you work out these and they roughly similar, can say model fits a POSSION DISTRUBTION

Question 10

Whatsa defining feature about possion compared to others in terms of n

Answer 10

N goes to INFINITY

Question 11

How to adjust position distribution if thr time period change?

Answer 11

Multiply the rate by the factor of time difference!

Question 12

Remmeber the fact that these are all DISCRETE variable distributions means what

Answer 12

It means that x can only be whole numbers, not like normal disturb ti on where it’s CONTINOUS instead

Question 13

How to add positions distinctions? What is the conditond

Answer 13

They Must be INDEPENDENT random variables x and y with means

Now x+y - po (mew + lambda)

Question 14

Can we subtrsvt position distrutbions? Why

Answer 14

NO

because x -y could give us negstive numbers

By defintion position distubtion goes form 0 to infinity, so can’t run

Question 15

When using a frequency table to find expected values for a POISSON distrubtion, how to do this

WHAT TO REMEMBER ABOUT FINDING FINAL PROBABILITY

Answer 15

Basically find the probability for each cagheroy and multiply by total to get expected frequency (because probability x total)

However remember the LAST category should be that and more, as possion goes to infinity

So need to find that probability a bit differently, DOMT lack , they should all add to 1

Question 16

What could be the reason why two sets of data, one follows posssion amd one doesnt

For example people entering a church / tourist shop

Answer 16

Think, random independent, uniform rate , singly

It might be that few are entering at the same time, so can’t say Idependenr as people are entering as a GROUP . Church visit in 10s but shop was Idependent as people coming in was small

• remember not modelling as grouo as it must be single, however people entering every at the same time are random, so in this case problem is that they aren’t indepent

Question 17

How to approximate a binomial distribution using position

What 2 CONDTIONS!

Answer 17

If X -B (n,p) then Y can be approximately - (NP)

Lambda is always the mean, and mean if binomial is np

BUT INLY IF N IS LARGE AND P IS SMALL

Question 18

Again what are two rules to approximating binomial as poisson

Answer 18

N is large

P is small

Question 19

Why would we want to even model binomial using possion what’s benefit

Hows this different to using normal to model binomial, why is this one more useful?

Answer 19

No benefit as you modelling a discrete distribution and another discrete distribution but it’s more the fact that we can could be useful

Normal more useful, you are modelling a CONTINOUS using a discrete . Need a Cl ti high correction for that , but yeah useful to allow calculators from longing it with factorials and CONTINOUS ti discrete has uses