Problems tagged with "maxium likelihood"
Problem #117
Tags: maxium likelihood
Consider Justin's right-triangle density. It is a parametric density with two parameters, \(\alpha\) and \(\beta\)(with \(\beta >0\)), and pdf:
A picture of the density is shown below for convenience:
![](../images/117/fig.png)
Your answers to the below problems should all be in the form of a number. You may leave your answer as an unsimplified fraction or a decimal, if you prefer.
Part 1)
Let \(\mathcal X = \{0, 1, 2\}\) be a data set of 4 points and let \(\mathcal L(\alpha, \beta; \mathcal X)\) be the likelihood function (with respect to this data). What is \(\mathcal L(0, 2)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood.
Part 2)
Let \(\mathcal X = \{0, 1, 2\}\) be the same data set as in the previous part. What is \(\mathcal L(0, 4)\)?
Part 3)
Now let \(\mathcal X = \{6, 7\}\) be a new data set of just two points. What are the maximum likelihood estimates of \(\alpha\) and \(\beta\) with respect to this data? Remember, \(\beta > 0\).
\(\alpha\): \(\beta\):