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[likelihood] fixed exercise

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Jan Benda 2021-01-11 21:41:32 +01:00
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likelihood/exercises/likelihood-1.tex
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@ -51,10 +51,10 @@ of the standard deviation.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\question \qt{Maximum-likelihood estimator of a line through the origin} \question \qt{Maximum-likelihood estimator of a line through the origin}
In the lecture we derived the following equation for an In the lecture we derived the following equation for an
maximum-likelihood estimate of the slope $\theta$ of a straight line maximum-likelihood estimate of the slope $m$ of a straight line
through the origin fitted to $n$ pairs of data values $(x_i|y_i)$ with through the origin fitted to $n$ pairs of data values $(x_i|y_i)$ with
standard deviation $\sigma_i$: standard deviation $\sigma_i$:
\[\theta = \frac{\sum_{i=1}^n \frac{x_i y_i}{\sigma_i^2}}{ \sum_{i=1}^n \[ m = \frac{\sum_{i=1}^n \frac{x_i y_i}{\sigma_i^2}}{ \sum_{i=1}^n
\frac{x_i^2}{\sigma_i^2}} \] \frac{x_i^2}{\sigma_i^2}} \]
\begin{parts} \begin{parts}
\part \label{mleslopefunc} Write a function that takes two vectors \part \label{mleslopefunc} Write a function that takes two vectors
@ -105,7 +105,7 @@ normally-distributed data. Such parameter need to be estimated
numerically by means of maximum-likelihood from the data. numerically by means of maximum-likelihood from the data.
Let us demonstrate this approach by means of data that are drawn from a Let us demonstrate this approach by means of data that are drawn from a
gamma distribution, gamma distribution.
\begin{parts} \begin{parts}
\part Find out which \code{matlab} function computes the \part Find out which \code{matlab} function computes the
probability-density function of the gamma distribution. probability-density function of the gamma distribution.
@ -123,7 +123,7 @@ gamma distribution,
\part Compute and plot a properly normalized histogram of these \part Compute and plot a properly normalized histogram of these
random numbers. random numbers.
\part Find out which \code{matlab} function fit a distribution to a \part Find out which \code{matlab} function fits a distribution to a
vector of random numbers according to the maximum-likelihood method. vector of random numbers according to the maximum-likelihood method.
How do you need to use this function in order to fit a gamma How do you need to use this function in order to fit a gamma
distribution to the data? distribution to the data?