Question
Assume that XL, Xn iid Normal(p = 0,02) , derive the maximum likelihood estimator of 02 NOTE: In this derivation you should: explicitly identify the likelihood function (in simplified form); explicitly identify the loglikelihood function (in a simplified form); 3 use the second derivative test to ensure that the estimator is indeed a maximum; and make sure you are differentiating with respect to 0? (and not with respect to o)_
Assume that XL, Xn iid Normal(p = 0,02) , derive the maximum likelihood estimator of 02 NOTE: In this derivation you should: explicitly identify the likelihood function (in simplified form); explicitly identify the loglikelihood function (in a simplified form); 3 use the second derivative test to ensure that the estimator is indeed a maximum; and make sure you are differentiating with respect to 0? (and not with respect to o)_
Answers
Refer to Exercise $3.86 .$ The maximum likelihood estimator for $p$ is $1 / Y$ (note that $Y$ is the geometric random variable, not a particular value of it). Derive $E(1 / Y) .[\text { Hint: If }|r|<1$, $\left.\sum_{i=1}^{\infty} r^{i} / i=-\ln (1-r) .\right]$.
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