TEMPORARY_DOCUMENT_ID Finding a numerical solution

Exercises

1.

A weighted six-sided dice has the probabilities of $\{1/4,~{}1/4,~{}1/6,~{}1/6,~{}1/12,~{}1/12\}$ for numbers 1 to 6 respectively. Write out the moment generating function for rolling this dice. Evaluate the expected value and variability of the weighted dice.
2.

A discrete random variable $X$ is said to have a geometric distribution with probability $\phi$ if it has the probability mass function (pmf):

$p_{X}(x)=\phi(1-\phi)^{x}\quad\mathrm{for}~{}x=0,1,2,\ldots,~{}~{}0<\phi<1.$

Show that the moment generating function (mgf) for the geometric distribution is:

$M(s)=\frac{\phi}{1-\exp\{s\}(1-\phi)}.$

For what values of $s$ is the mgf valid.
3.

A uniform random variable $Y$ on the unit interval has the probability density function (pdf):

$f_{Y}(y)=\left\{\begin{array}[]{ll}1,&\mathrm{if}~{}0<y<1,\\ 0,&\mathrm{otherwise}\end{array}\right.$
1. (a)
  
  Derive the mgf $M(s)$ and cumulant generating function (cgf) $K(s)$ for this uniform distribution.
2. (b)
  
  Show that the exponential family of distributions generated from the uniform distribution is:
  
  $f_{Y}(y|\theta)=\left\{\begin{array}[]{ll}\frac{\theta\exp\{y\theta\}}{\exp\{% \theta\}-1},&\mathrm{if}~{}0<y<1,\\ 0,&\mathrm{otherwise}\end{array}\right.$
3. (c)
  
  Plot this pdf for $\theta=-2$ , $\theta=-1$ , $\theta=0$ and $\theta=1$ . Describe the shape of these distributions.
4.

Consider the binomial distribution with known size $n$ and unknown probability $\pi$ .
1. (a)
  
  Show that this belongs to the exponential family with canonical parameter $\theta=\log\left(\frac{\pi}{1-\pi}\right)$ .
2. (b)
  
  Find the mean function in terms of the canonical parameter, i.e. derive $\mu=m(\theta)$ .
3. (c)
  
  Show that the mean function is monotonic, i.e. a one-to-one function, and derive the inverse function $\theta=m^{-1}(\mu)$ .
4. (d)
  
  Derive an expression for the variance function $v(\mu)$ .
5.

A psychologist is interested in examining how children interpret simple instructions. The experiment involves asking children to make a mark on a line of unit length according to her instructions.

The first instruction is to ”make a mark near to the right-hand end of the line”. It is assumed that the distribution of the marks follow a Beta( $\alpha$ , 1) distribution with pdf:

$f_{Y}(y)=\alpha y^{\alpha-1}$
1. (a)
  
  Show that this distribution belongs to the exponential family with canonical parameter $\theta=\alpha-1$ and sufficient statistic $t(y)=\log(y)$ . What happens to the pdf when $\theta=0$ ?
2. (b)
  
  Let $y_{1},\ldots,y_{n}$ be distance of the mark from the left-hand edge of the line from $n$ randomly selected children. Derive an expression for the maximum likelihood estimate $\hat{\theta}$ and expected information at the MLE.
3. (c)
  
  The table below presents the fraction of the line from the left-hand end to the mark by 10 randomly selected children. Calculate an approximate 95% interval for the canonical parameter. What can you conclude about the children’s understanding of the instruction.
  
  0.82 0.95 0.98 0.66 0.98
  
  0.82 0.97 0.85 0.73 0.97

0.82	0.95	0.98	0.66	0.98
0.82	0.97	0.85	0.73	0.97