Workshop questions

Complete the random variable selection flowchart that is available on the Moodle site.

Let $Z\sim \text{Gamma}(\alpha ,\beta )$. Show that $𝖤[X]=\alpha /\beta$ and $\mathrm{𝗏𝖺𝗋}(X)=\alpha /{\beta }^2$.

A bag contains four dice labelled 1,2,3,4. The die labelled $j$ has $j$ white faces and $(6-j)$ black faces. A die is chosen at random from the bag and rolled. Define the random variables $X$ and $Y$ as follows:

1. (a)

   Construct a table displaying the (marginal) probability mass function for $X$ and a separate table displaying the joint probability mass function for $X$ and $Y$.

2. (b)

   Use the definition of conditional probability to find $𝖯(X=2|Y=1)$.

Let $T_1$ and $T_2$ be independent continuous random variables with $T_1\sim \text{Exponential}(\lambda )$ and $T_2\sim \text{Exponential}(\mu )$. Let $T=\min (T_1,T_2)$. Express the event $\{T>t\}$ as an intersection of two events involving $T_1$ and $T_2$. Hence find $𝖯(T>t)$. What is the distribution of $T$?

A fair die is thrown $n$ times

1. (a)

   Let $X$ be the score on the first throw. Calculate $𝖤[X]$ and $\mathrm{𝗏𝖺𝗋}(X)$.

2. (b)

   Let $S$ be the sum of scores obtained on all $n$ throws. By representing $S$ as the sum of suitable random variables find $𝖤[S]$ and $\mathrm{𝗏𝖺𝗋}(S)$.

3. (c)

   Calculate $𝖤[S/n]$ and $\mathrm{𝗏𝖺𝗋}(S/n)$. Compare these with your answers to part (i). Use this to explain why scientists average the result of several replicated experiments.