MATH235 Week 7 - Workshop problems

Take logs of $f(x)$ and simplify the result in the following cases:

1. (a)

2. (b)

   $f(x)=\lambda \exp (-\lambda x)\text{ where }\lambda >0$ v $f(x)={(\beta x)}^y\text{ where }\beta >0$

3. (c)

   $f(x)=\theta x\exp \left(-\theta x^2\right)\text{ where }\theta >0$

4. (d)

   $f(x)={\prod }_{i=1}^n{\lambda }^{x_i}\exp (-\lambda )\text{ where }\lambda >0\text{ and }\forall i,x_i>0.$

1. (a)

   Obtain the first and second derivatives with respect to $\theta$ of

   • (i)

   • (ii)

     $-\frac{1}{2}{(x-\theta )}^2$

   • (iii)

     $-\frac{1}{2}\log \theta -\frac{1}{2}\frac{x^2}{\theta }\text{ where }\theta >0$

2. (b)

   Obtain the first and second derivatives of the following with respect to $\beta$, simplifying your answer as much as you can:

   • (iv)

     ${\sum }_{i=1}^n{(y_i-\beta )}^2$

   • (v)

     $\log {\prod }_{i=1}^n{(\beta x_i)}^{y_i}\mathit{\hspace{1em}\hspace{1em} }\text{where}\mathit{\hspace{1em}\hspace{1em} }\beta >0\mathit{\hspace{1em}}\text{and}\mathit{\hspace{1em}\hspace{1em} }\forall i,x_i>0\mathit{\hspace{1em}}\text{and}\mathit{\hspace{1em}}{y}_i>0.$

Suppose that $X$ is a discrete random variable with the following probability mass function:

where $0\le \theta \le 1$ is a parameter of interest. Suppose a sample of $x=(x_1,\mathrm{\dots },x_n)$ is observed, and let $n_i$ ($i=0,\mathrm{\dots },3$) represent the number of times data value $i$ appears in the sample, so that ${\sum }_{i=0}^3{n}_i=n$.

1. (a)

   Derive the method of moments estimator for the parameter $\theta$ based on the sample $x$.

2. (b)

   Write down the likelihood and log-likelihood functions for the parameter $\theta$ based on the sample $x$.

3. (c)

   Find an expression for the maximum likelihood estimator for $\theta$ and verify it is indeed a maximum.

In the 2010-11 season, Wolverhampton Wanderers played 38 games, winning 11, drawing 7 and losing 20, finishing the season with 40 points (just escaping relegation!). Let $\theta$ be the probability that the team wins a randomly selected match.

1. (a)

   It is proposed that the number of games won by Wolves is modelled as a binomial distribution with parameter $\theta$. Discuss the assumptions this would be making, and whether they are reasonable.

2. (b)

   Write down the resulting likelihood and log-likelihood functions.

3. (c)

   Find the maximum likelihood estimate of $\theta$.

4. (d)

   At one point in the season, Wolves went for 8 games without a win. Is this evidence against the binomial distribution being appropriate?