sir this is question i translated need help

and this is translation:

The Residents-Count of a german Town need to be described with a fitting density function. For this only towns with over 1000 Residents should be counted. Use the following density function

...... density function is here

Using samples X1, ...., Xn from Independent identical ( using above density function) distributed Residents-Count, specify the density function further.

a) calculate the Maximum-Likelihood-Appromitation for the parameter ? for the samples X1, ..., Xn.

Hint: The proof that it is a global maximum is not required.

b) Give a asymptotic distribution for the Appromitation in a)

Now these samples are found:

1.3, 3.7, 7.4, 2.8, 2.2

which describe the Population (in Thausands)

c) Calculate using the samples the in a) calculated ML-Estimator.

Hint: If no solution found in a) use the following ML-Estimator......

d) Plot the Normalized Log-Likelihood function for the Samples with Markings and Description.

e) Calculate the 99% Wald-Konfidenz Intervall for the Parameter ? for the Samples. Comment on your solution and describe why it is sensible.

Hint: If no solution in a) and b) are found, use these.......

Chosen p-Quantile zp from the Standard normal distribution

z0.9 = 1.28 ; z0.95 = 1.64 ; z0.975 = 1.96 ; z0.99 = 2.33 ; z0.995 = 2.58

f) Finally check if the above density function specified distribution is a suitable model for the distribution of the Residents-Count. The Quantile are calculated:

........

You can do ? 2 -Test of model adaption for a level of significance of ? = 5% and specify the solution. Why is the use of a ? 2 -Test here critical ?

Hint:

In the lectures the ? 2 -Test was only described for discrete random variables. To apply the Test here, the steady distribution here need to be converted to discrete by giving the carrier fitting intervals.

Chosen p-Quantile of the ? 2 -Test distribution with k freedom-degrees:

and table here

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