5. Suppose there are n trials in a binomial experiment, but 0 is unknown. Let X be the random variable with this distribution, and suppose that our experiment gave us zo successes. (a) Calculate P(X = xo) in terms of 0. We interpret this as the likelihood of getting the observed result under a particular choice of 0.

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5. Suppose there are n trials in a binomial experiment, but 0 is unknown. Let X be the random
variable with this distribution, and suppose that our experiment gave us ro successes.
(a) Calculate P(X = xo) in terms of 0. We interpret this as the likelihood of getting the observed
result under a particular choice of 0.
(b) Show that the likelihood is maximized when 0 = m. In other words, a is our best guess for
what 0 might be, given the evidence we have from our experiment.
(c) Based on the previous item, the random variable is the maximum likelihood estimator for
0. Calculate E().
(d) Calculate V(). What happens to the variance as n gets large?
Transcribed Image Text:5. Suppose there are n trials in a binomial experiment, but 0 is unknown. Let X be the random variable with this distribution, and suppose that our experiment gave us ro successes. (a) Calculate P(X = xo) in terms of 0. We interpret this as the likelihood of getting the observed result under a particular choice of 0. (b) Show that the likelihood is maximized when 0 = m. In other words, a is our best guess for what 0 might be, given the evidence we have from our experiment. (c) Based on the previous item, the random variable is the maximum likelihood estimator for 0. Calculate E(). (d) Calculate V(). What happens to the variance as n gets large?
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