Problems

When I throw a dart at a target, the probability that it lands a distance \(X\) from the centre is a random variable with density function \[ \mathrm{f}(x)=\begin{cases} 2x & \text{ if }0\leqslant x\leqslant1;\\ 0 & \text{ otherwise.} \end{cases} \] I score points according to the position of the dart as follows: %

%\newline\hspace*{10mm} if~\(0\le X< \frac14\), my score is 4; %\newline\hspace*{10mm} if~\(\frac14\le X< \frac12\), my score is 3; %\newline\hspace*{10mm} if \(\frac12\le X< \frac34\), my score is 2; %\newline\hspace*{10mm} if \(\frac34\le X\le 1\), my score is 1.

1. Show that my expected score from one dart is 15/8.
2. I play a game with the following rules. I start off with a total score 0, and each time~I throw a dart my score on that throw is added to my total. Then: \newline \hspace*{10mm} if my new total is greater than 3, I have lost and the game ends; \newline \hspace*{10mm} if my new total is 3, I have won and the game ends; \newline \hspace*{10mm} if my new total is less than 3, I throw again. Show that, if I have won such a game, the probability that I threw the dart three times is 343/2231.

The cakes in our canteen each contain exactly four currants, each currant being randomly placed in the cake. I take a proportion \(X\) of a cake where \(X\) is a random variable with density function \[{\mathrm f}(x)=Ax\] for \(0\leqslant x\leqslant 1\) where \(A\) is a constant.

1. What is the expected number of currants in my portion?
2. If I find all four currants in my portion, what is the probability that I took more than half the cake?

The maximum height \(X\) of flood water each year on a certain river is a random variable with density function \begin{equation*} {\mathrm f}(x)= \begin{cases} \exp(-x)&\text{if \(x\geqslant 0\),}\\ 0&\text{otherwise}. \end{cases} \end{equation*} It costs \(y\) megadollars each year to prepare for flood water of height \(y\) or less. If \(X\leqslant y\) no further costs are incurred but if \(X\geqslant y\) the cost of flood damage is \(r+s(X-y)\) megadollars where \(r,s>0\). The total cost \(T\) megadollars is thus given by \begin{equation*} T= \begin{cases} y&\text{if \(X\leqslant y\)},\\ y+r+s(X-y)&\text{if \(X>y\)}. \end{cases} \end{equation*} Show that we can minimise the expected total cost by taking \[y=\ln(r+s).\]