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:

\[ f(x ; \alpha, \beta) = \begin{cases}{\displaystyle \frac{2}{\beta} - \frac{2(x - \alpha)}{\beta^2} }, & \text{if } x \in[\alpha, \alpha + \beta] \\ 0, & \text{otherwise} \end{cases}\]

A picture of the density is shown below for convenience:

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, \(\beta > 0\).

\(\alpha\): \(\beta\):