MATH235 Week 1 - Assessed problems (coursework)

Consider the sequence of random variables $X_1,\mathrm{\dots },X_n$. Assume that these are IID, with Poisson$(\mu )$ distribution, $\mu >0$.

1. (a)

   Show that the sample mean

   $${\overline{X}}_n=\frac{X_1+\mathrm{\dots }+X_n}{n}$$

   is an unbiased estimator of $\mu$.

   [marks: 1]

2. (b)

   What is the variance of the estimator ${\overline{X}}_n$?

   [marks: 2]

3. (c)

   Now suppose that $X_1,\mathrm{\dots },X_n$ are independent, but that for $i=1,\mathrm{\dots },n$,

   $$X_i\sim \text{Poisson}\left(w_i\lambda \right),w_i>0,\lambda >0,$$

   for some weights $w_1,\mathrm{\dots },w_n$.

   The bias of an estimator $\widehat{\theta }$ for a parameter $\theta$ is the difference between the expected value of the estimator and the parameter,

   $$𝔼[\widehat{\theta }]-\theta .$$

   Calculate the bias if the sample mean ${\overline{X}}_n$ is used as an estimator for $\lambda$. What constraint can be placed on the weights to ensure the estimator is unbiased?

   [marks: 2]

Using the data in Table 0.3 we wish to test the hypothesis that the mean price of tea is greater than 1.8 dollars per kg. Assume that the tea prices are an IID sample from a Normal$(\mu ,{\sigma }^2)$ distribution, with unknown population variance.

1. (a)

   Write down appropriate null and alternative hypotheses for this test.

   [marks: 1]

2. (b)

   Calculate an appropriate test statistic for this test.

   [marks: 2]

3. (c)

   Calculate the $p$-value for the test statistic obtained in part (b). What conclusions can you draw about the annual price of tea?

   [marks: 2]

Challenge
Suppose that $X_1,\mathrm{\dots },X_n$ is an IID sequence of Binomial$(m,p)$ random variables, and that $Y_1,\mathrm{\dots },Y_n$ is also a sequence of Binomial$(m,p)$ random variables. Further the correlation of the two sequences is given by

Consider the estimator $\widehat{p}=\frac{1}{2m}(\overline{X}+\overline{Y})$.

1. (a)

   Show that $\widehat{p}$ is an unbiased estimator of $p$.

   [marks: 2]

2. (b)

   Express the variance of $X_i+Y_i$ in terms of $m$, $p$ and $r$.

   [marks: 1]

3. (c)

   Using your answer to parts $(i)$ and $(ii)$, give an expression for the variance of $\widehat{p}$.

   [marks: 2]