Math for Quantitative Finance
# Confidence and Estimation

One type of parameter estimation is maximum likelihood estimation. The MLE of a distribution given some data has an intuitive definition: it is the parameter(s) for the model that would make observing the given data most likely.

What is the MLE for the success probability of a geometric distribution which takes an observed \(n=4\) trials to achieve the first success?

You have a coin which you know lands on heads with probability \(P.\) You believe that P is normally distributed with mean 0.5 and variance 0.01. If it is flipped 10 times and there are 6 heads, which is the best estimate for \(P?\)

Technically speaking, we're looking for a "maximum a posteriori probability" (MAP). It's like an MLE, except that we Bayesian update from a non-uniform prior distribution for the parameter.

An airline has numbered their planes \(1,2,\ldots,N,\) and you observe the following 3 planes, which are randomly sampled from the \(N\) planes:

What is the maximum likelihood estimate for \(N?\) In other words, what value of \(N\) would, according to conditional probability, make your observation most likely?

**unbiased** estimator: one which has an expected value equal to the true value of the parameter.

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