# Foundation Statistics Mod

#### Jon123

Hello everyone!

I am having trouble getting in contact with my lecturer so I am hoping for assistance here with a few questions and would like support in answering them.
The set data given are arrival times e.g. 11.59pm, 11.43pm,11.23pm ect. all on the same day.
1. Draw the histogram of the Arrival Distribution and use the 5% significance test to test whether it is coming from an exponential distribution.
Do I group the data into interval times?
How to solve exponential. Do I use f(x) = 1/theta . exp ^ -x/theta

2. Estimate the parameter of the distribution
3. Show that the estimate used to estimate its parameter is the best estimate
4. Suppose the population increase by 100% within the next 10 years give the equation of the predicted distribution and draw its histogram
5. If every 5th customer is getting a voucher give the arrival distribution of the customers who will receive a voucher
6. Give the distribution of the number of customers arriving in every half an hour

#### chiro

MHF Helper
Hey Jon123.

Can you show us what you have tried? To start you off, for number 2, consider the likelihood with the sample (given n observations) and then for MLE solve for the log-likelihood where its derivative is zero (I assume you want to use MLE but if not just show your attempt for the other estimator).

#### Jon123

Hi Chiro!
f(x,θ) = (1/θ) * exp(-x/θ) 0 ≤ x < ∞
this will equal to (1/θ)^n * exp^ -Σ(-x/θ)
Then solve for θ, which it will equal to θ = Σx/n = x bar
To solve moment est E(x) = integral ∞,0 x * (exp(-x/θ) / θ) dx = θ ect..
m.l.e is unbiased for θ .. so, E(x bar) = θ
E(x1+...+xn / n) = nE(x)/n = E(x) = θ therefore x bar is unbiased for θ
Using Rao cramer lower bound(rclb)
V(x bar) =θ^2 / n ... will equal -1/θ^2 = -1/n[1/-1/θ^2] = θ^2/n .. therefore x bar is the best estimator.

I dont quite understand questions 4, 5 and 6
Are you able to help me answer 2 and 3 ? from what I did above should be correct in theorem. But do i need to replace θ with a number...
The data that given where times from am to pm.. n=107 entries..

#### chiro

MHF Helper
For number 4, consider what happens to the rate when you double the population: if the you double the amount of people that come through the gate then what would this do to the average rate for the exponential?

For number 5, you need to consider a conditional distribution: you have every 5th person get a voucher so consider a random sample of five people are taken then consider P(Next-Person-Gets-Voucher|Last-Multiple-Of-Five-Got-A-Voucher) in which conditional probabilities are given by P(A|B) = P(A and B)/P(B).

For number 6, consider what happens to the rate when you are going a "per hour" rate to a "per half hour" rate: How is the rate adjusted in the new set of units?