The probability density function (p.d.f.) of a continuous random variable $X$ is defined to be:

$f(x) = \begin{cases} \frac{x}{6} + k \text{ for } 0 < x < 3 \\ 0 \text{ otherwise,} \end{cases}$
for some constant k. For these problems, please ensure your answers are accurate to within 3 decimals.

Part a)

Find the value of k that makes the above function a proper p.d.f.



Part b)

Hence find $P(0.5 < X < 1).$