Homework Problem 15, Due Monday March 8.
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Consider the "dataset" consisting of 

n_1 = 100 obs with   XV = 1  resulting in m_1 = 19 "successes"
n_2 =  50 obs with   XV = 2  resulting in m_2 = 13 "successes"
n_3 = 120 obs with   XV = 3  resulting in m_3 = 30 "successes"
n_4 =  80 obs with   XV = 4  resulting in m_4 = 24 "successes"

Maximize the likelihood for these data under the model
that the trials with  XV = k  are iid with success 
probability  p_k = 0.1*log(1+b*k), k=1,...,4, where  b  
is an unknown positive parameter. 

Verify as clearly as possible that the maximum likelihood 
estimator you find is the global maximum, using either 
purely numerical techniques OR the hint that the data were 
actually simulated from the indicated model with some 
"true" b.

HINTS:  (1) You can solve for b in terms of any ONE of 
the probabilities, say p_k , by  b = (exp(10*p_k)-1)/k . 
Thus a reasonable starting value for the likelihood 
maximization (using k=3) would be 
    (exp(10*m_3/n_3)-1)/k = 3.7275

(2) Also as mentioned in class: in terms of the true value 
b_0  the quantities  
(m_k-n_k p_k(b_0))^2/(n_k p_k(b_0) (1-p_k(b_0))) 
behave approximately as independent chi-square variables.