This write up is about an estimator. A statistical estimator, is used when you already have a model in mind, data at hand and want to estimate some parameters needed for the model. For example, you want to predict how many runs a batter would make in a game given recent history \(x_1,x_2,x_3,\ldots\) . If we assume (we are making a choice of a model now) that the scores come from a normal distribution with mean \(\mu\) and standard deviation \(\sigma\) then the probability density function for a given value \(x\) is

The likelihood that a series of points \(x_1,x_2,x_3,\ldots\) come from such a distribution can be expressed as

Next is basic calculus. Take the logarithm on both sides, set the partial derivative w.r.t. \(\mu\) to zero yields (excluding the algebra)

To verify, you also need to check the second derivative's sign to see if its negative to ensure that it is indeed a maxima you have found.

So a simple estimator would be to use the average runs scored from the past \(n\) days.…

The likelihood that a series of points \(x_1,x_2,x_3,\ldots\) come from such a distribution can be expressed as

Next is basic calculus. Take the logarithm on both sides, set the partial derivative w.r.t. \(\mu\) to zero yields (excluding the algebra)

To verify, you also need to check the second derivative's sign to see if its negative to ensure that it is indeed a maxima you have found.

So a simple estimator would be to use the average runs scored from the past \(n\) days.…