HOMEWORK 1 HINTS - Last modified: Sunday Jan 28, 2007, 5 p.m.
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Problem 8: you may follow the strategy shown in Case Study 5.2.1 of page 352. Note that k represents the variable, which in this case is discrete (similar to the "y" variable in continuous cases). The "expected frequency" column of the table is what you produce *after* you have computed the estimate for the parameter, and then use it to produce the "expected frequency" column (this is the last thing you do).

Problem 12:
Just think of "k" as a known constant. (to get started, you may set k to 1 if that helps, and then do the case with general k later). So, L = Product[ t k^t (1/y_i)^(t+1) , i=1..25]
here I use "t" for "theta".
take logarithm and take derivative with respect to "t".
Use properties of logarithms to get a sum.
After taking derivatives and setting = 0 you may solve for t.

Problem 22:
In this problem, the method of moments gives 2 equations.
The left-hand side of the first equation is just "mu",
since we already know from mth451 that the mean is mu.
Then one concludes that mu_e = 1/n (Sum[y_i,i=1..n]).
The second equation is E[Y^2] = 1/n (Sum[ y_i^2,i=1..n]).
But we know from mth451 that sigma^2 = E[(Y-mu)^2] = E[Y^2]-(E[Y])^2.
From this and previous equations we get the answer for sigma^2.

Problem 24.
Here we need the moment equations for the case of a discrete probability function. They are given in page 358, line number 7.
Actually you need only one equation (for j=1)
since there is only one parameter for the geometric distribution.
The Left-hand-side is the expected value
of X. So the equation is,
E[X] = 1/n Sum[k_j , j=1..n] But here you may use the fact that for geometric distribution, E[X] = 1/p,
according to page 318 Theorem 4.4.1.
So plug this into the previous equation to solve for p_e.
Finally, the "expected frequencies" are found in a way similar to the example at the top of page 353: plug the parameter value p_e into the formula for the probability function, and then plug in k=1,2,3,..8, multiply by 250 to get an additional column in the table.