1. Suppose that Y; takes values 0, 1, 2 with probability f(y;0) = e®y-b(@). (a) Find the function b(0) so that f(y; 0) is a probability mass function (i.e., total probability one). Find the values f(0; 0), f(1;0), and f(2;0) for 0 = 0, 0 = log .5, 0 = log 2 (remember that unless otherwise specified, all logs are natural logs). Find the expected value of Y for each of these three values of 0. (b) If we observe Y1,Y2, ..., Yn, find the log-likelihood function ((0 ;Y1,Y2, ...,Y,). (c) Suppose that we have a sample of size 50 and observe the following summary of values of Y;: Y = 0 Y =1 Y = 2 Total 26 11 13 50 (I.e., 11 of the Y; have value 0, etc.) Find the mean of the Y; and compare to the means from part (a). What does this tell you about the MLE for 0?
1. Suppose that Y; takes values 0, 1, 2 with probability f(y;0) = e®y-b(@). (a) Find the function b(0) so that f(y; 0) is a probability mass function (i.e., total probability one). Find the values f(0; 0), f(1;0), and f(2;0) for 0 = 0, 0 = log .5, 0 = log 2 (remember that unless otherwise specified, all logs are natural logs). Find the expected value of Y for each of these three values of 0. (b) If we observe Y1,Y2, ..., Yn, find the log-likelihood function ((0 ;Y1,Y2, ...,Y,). (c) Suppose that we have a sample of size 50 and observe the following summary of values of Y;: Y = 0 Y =1 Y = 2 Total 26 11 13 50 (I.e., 11 of the Y; have value 0, etc.) Find the mean of the Y; and compare to the means from part (a). What does this tell you about the MLE for 0?
Chapter6: Exponential And Logarithmic Functions
Section6.8: Fitting Exponential Models To Data
Problem 56SE: Recall that the general form of a logistic equation for a population is given by P(t)=c1+aebt , such...
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