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The continuous random variable \(X\) has the following cumulative distribution function: $$\text{F}(x) = \begin{cases} 0, & x < 0, \\ \frac{1}{432} x^2(x^2 - 16x + 72), & 0 \leq x \leq 6, \\ 1, & x > 6. \end{cases}$$

1. Find P(\(X < 2\)). [2 marks]
2. Find and specify fully the probability density function f(\(x\)) of \(X\). [4 marks]
3. Show that the mode of \(X\) is 2. [6 marks]
4. State, with a reason, whether the median of \(X\) is higher or lower than the mode of \(X\). [1 mark]

The length of time, in tens of minutes, that patients spend waiting at a doctor's surgery is modelled by the continuous random variable \(T\), with the following cumulative distribution function: $$F(t) = \begin{cases} 0, & t < 0, \\ \frac{1}{135}(54t + 9t^2 - 4t^3), & 0 \leq t \leq 3, \\ 1, & t > 3. \end{cases}$$

1. Find the probability that a patient waits for more than 20 minutes. [3]
2. Show that the median waiting time is between 11 and 12 minutes. [3]
3. Define fully the probability density function f(t) of \(T\). [3]
4. Find the modal waiting time in minutes. [4]
5. Give one reason why this model may need to be refined. [1]

The continuous random variable \(X\) has cumulative distribution function given by $$F(x) = \begin{cases} 0 & x \leq 0 \\ k\left(x^3 - \frac{3}{8}x^4\right) & 0 < x \leq 2 \\ 1 & x > 2 \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = \frac{1}{2}\) [1]
2. Showing your working clearly, use calculus to find
   1. E(\(X\))
   2. the mode of \(X\)
   [6]