Problem:
1. The number of messages left on your answering machine during 14 successive days are 2, 1, 1, 3, 1, 0, 0, 3, 0, 1, 4, 0, 0, 2..
Select a model for this problem.
Find the maximum likelihood estimate of the average number of messages left per day on your answering machine in this model for general n and general data.
Compute the MLE for the above data.
2. Let X1,X2,....,Xn be a random sample from the following distributions, find the maximum likelihood estimate of θ in each case.
f(xlθ)= θ(1-x)^(θ-1), 0≤x≤1, and zero elsewhere, θ>0.
f(xlθ)=θx^(θ-1), 0≤x≤1, and zero elsewhere, θ>0.
P(X=xlθ)= θ(1-x)^(x-1),x=1,2,.......,0<θ<1.
3. Let X1,X2,....,Xn be a random sample from an exponential distribution with mean 1/θ, θ>0. Find the maximum likelihood estimate of 1/θ, and show that it is consistent and asymptotically normal. What is the maximum likelihood estimate of θ?
4. Let X1,X2,....,Xn be a random sample from a Poisson distribution with parameter λ. Find the maximum likelihood estimate of P( X=0)=e^(-λ).
5. For the fifth problem see the attachment.
Attachment:- refer.rar