Question

Suppose Craig wants to redistribute officers among precincts so as to reduce the maximum amount of time callers in any one precinct have to wait for a police response. What should he do and what impact would this have?

Answer 1

Firstly, Craig have to use binomial distribution to distribute officers among precincts . Using combinations and permutation he can allocate the respective call to the respective officer. This will help develop a network, using various samples of different combinations can help him reach he outcome. Find the probability of each outcome then find the pdf of binomial distribution to get the idea. Now find Minimum Variance Unbiased Estimator (MVUE). Use the binomial pdf to get the MVUE and choose the result with minimum variance and then compare all. Then he can use the Maximum Likelihood Estimator (MLE).

Formula for binomial distribution is-

B(x,n,p)= nCx*P^x*(1-P)^n-x   for x=0,1,2,…..n.

Mean(µ) = np
Variance(σ²) = npq

• The variance of a Binomial Variable is always less than its mean. ∴ npq<np.
• For Maximum Variance: p=q=0.5 and σ_max = n/4.

An estimator φˆ of a parameter φ = φ(θ) is Uniformly Minimum Variance Unbiased (UMVU) if, whenever φ˜ is an unbiased estimate of φ we have Varθ(φˆ) ≤ Varθ(φ˜) We call φˆ the UMVUE. (‘E’ is for Estimator.) The point of having φ(θ) is to study problems like estimating µ when you have two parameters like µ and σ .

Finally, The goodness of fit of a statistical model describes how well it fits a set of observations. Measures of goodness of fit typically summarize the discrepancy between observed values and the values expected under the model in question. So , this will help provide a clear picture about the distribution of data.

The above process will help reduce the maximum time of callers in any one precinct and each call will be distributed uniformly through each officers. Hence the efficiency will increase.