For binomial response data that follow the binomial distribution, the log-likelihood function for one observation (y1) is

= C + k ln(Ï€1) + (n1 €“ k) ln(1 €“ Ï€1)(7.32)
Where C is a constant that does not influence Ï€1, we can drop C from the above equation without any impact on the maximum likelihood estimate. Assume that you observed y1 = 5 malignant cells out of a sample of n1 = 12. Substituting these values into Equation (7.32) and dropping C simplifies the log-likelihood function to ln[ P(Y1 = 5)] = 5 ln(Ï€1) + 7 ln(1 €“ Ï€1).
a. Plot the log-likelihood function for several values of π1 between 0 and 1. Use the plot to estimate the value of π1 that will maximize the log-likelihood function.
b. If you have had calculus, set the derivative of the log-likelihood function to 0 and solve for π1. What is the maximum likelihood estimate for π1?

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In[PCX, = k)] = In)-ta - m