## Question

###### 11. (12 points) Waiting time plus service time in a telephone queue is modeled as a...

11. (12 points) Waiting time plus service time in a telephone queue is modeled as a gamma distribution with density fx (x/a) = { d?ze-de for = 20 for 1 <0. (a) Give the likelihood function based on n observations. (b) Find the maximum likelihood estimator for A. (c) Determine Å for observations 0.626, 0.918, 0.861, 0.455, and 1.648 minutes.

## Answers

coiren fxi(l2)= A2 eien O XLO u] Likelihood function, L (AL IT Exf wildl 2 -dxi = xi e Lal 20 Tixi & Taxi xco for maximum likell hood estima toe b] Toking log on both side sof : log L (N/= 20 lugnt tog Txi - Edxi DiPro writ. d a logL (W1 = 20 + - Exi a logL(A) - o Consides 22 log LC a = - 20 d2 20 - Exi = 0 mann d Hence 20 înen is maximun likelihood estimate .. 25 XiN Tour را به ] Gixen, 0.626 816.0 0.861 xar 0.455 X5 = 10.648 II i - [ 0.6207 648 6264 0.918 +0.861 +0.45 +1.648 ul- 4.508 5 0.9016 amel با اله 11 9106.0 2.2 1827 Amen 2.2 183 -Cons ,

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