A2 October 2020 Q5

EdexcelCurrent spec10 marksContinuous Random Variables

5.

Figure 1: sketch of y = f(x), a smooth bell-shaped curve rising from O, reaching a maximum at the middle and returning to the x-axis at 2π
Figure 1

The random variable \(X\) has probability density function \(\mathrm{f}(x)\) and Figure 1 shows a sketch of \(\mathrm{f}(x)\) where

\[\mathrm{f}(x) = \begin{cases} k(1 - \cos x) & 0 \leqslant x \leqslant 2\pi \\ 0 & \text{otherwise} \end{cases}\]
(a) Show that \(k = \dfrac{1}{2\pi}\) (3)

The random variable \(Y \sim \mathrm{N}(\mu, \sigma^2)\) and \(\mathrm{E}(Y) = \mathrm{E}(X)\)

The probability density function of \(Y\) is \(\mathrm{g}(y)\), where

\[\mathrm{g}(y) = \frac{1}{\sigma\sqrt{2\pi}}\mathrm{e}^{-\frac{1}{2}\left(\frac{y-\mu}{\sigma}\right)^2} \qquad -\infty \lt y \lt \infty\]

Given that \(\mathrm{g}(\mu) = \mathrm{f}(\mu)\)

(b) find the exact value of \(\sigma\) (3)
(c) Calculate the error in using \(\mathrm{P}\left(\dfrac{\pi}{2} \lt Y \lt \dfrac{3\pi}{2}\right)\) as an approximation to \(\mathrm{P}\left(\dfrac{\pi}{2} \lt X \lt \dfrac{3\pi}{2}\right)\) (4)
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