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 \(\mathrm { P } \left( X < \frac { 2 - \mu } { \sigma } \right)\).
4 (a) A random variable $X$ has probability density function defined by
$$\mathrm { f } ( x ) = \begin{cases} k & a < x < b \\ 0 & \text { otherwise } \end{cases}$$
(i) Show that $k = \frac { 1 } { b - a }$.\\
(ii) Prove, using integration, that $\mathrm { E } ( X ) = \frac { 1 } { 2 } ( a + b )$.\\
(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:
$$f ( x ) = \begin{cases} k & - 2 < x < 4 \\ 0 & \text { otherwise } \end{cases}$$
(i) Write down the value of the mean, $\mu$, of $X$.\\
(ii) Evaluate the standard deviation, $\sigma$, of $X$.\\
(iii) Hence find $\mathrm { P } \left( X < \frac { 2 - \mu } { \sigma } \right)$.
\hfill \mbox{\textit{AQA S2 Q4}}