A random variable \(X\) has probability density function defined by
$$\mathrm { f } ( x ) = \begin{cases} k & a < x < b 0 & \text { otherwise } \end{cases}$$
Show that \(k = \frac { 1 } { b - a }\).
Prove, using integration, that \(\mathrm { E } ( X ) = \frac { 1 } { 2 } ( a + b )\).
The error, \(X\) grams, made when a shopkeeper weighs out loose sweets can be modelled by a rectangular distribution with the following probability density function:
$$f ( x ) = \begin{cases} k & - 2 < x < 4 0 & \text { otherwise } \end{cases}$$
Write down the value of the mean, \(\mu\), of \(X\).
Evaluate the standard deviation, \(\sigma\), of \(X\).
Hence find \(\mathrm { P } \left( X < \frac { 2 - \mu } { \sigma } \right)\).