Calculate the expected value, 𝐸 (𝑋) = 𝜇 = ∫ 𝑥𝑝(𝑥) 𝑑𝑥 ∞ −∞, of a continuous random variable where 𝑝(𝑥) is the probability density function.

A continuous random variable \( X \) has probability density function \[ p(x) = \begin{cases} kx(4-x) & 0 \leq x \leq 4 \\ 0 & \text{otherwise} \end{cases} \] where \( k \) is a constant. The expected value \( E(X) \) is

A manufacturing process produces metal rods whose lengths \(L\) (in metres) follow a continuous probability density function given by \[p(\ell) = \begin{cases} k(3\ell - \ell^2) & \text{for } 0 \leq \ell \leq 3 \\ 0 & \text{otherwise} \end{cases}\] where \(k\) is a constant. (a) Show that \(k = \dfrac{2}{9}\). [2 marks] (b) Calculate the expected length of a rod produced by this process. Give your answer correct to two decimal places. [3 marks]

The time (in hours) that a customer spends at a retail outlet is modelled by a continuous random variable $X$ with probability density function $$p(x) = \begin{cases} k(6x - x^2) & \text{if } 0 \le x \le 6 \\ 0 & \text{otherwise} \end{cases}$$ Calculate the expected time a customer spends at the outlet.