Enrichment Practice: Continuous Random Variables

Outside current Core 9758 scope

This exercise develops ideas that support later probability study, but it is not presented as a prediction of a 9758 examination question.

Answers: Continuous Random Variables Answers.

A continuous random variable $X$ has probability density function

$$f(x)=\{\begin{array}{ll}kx,& 0\le x\le 2,\\ 0,& \text{otherwise}.\end{array}$$

(a) Use the total area under a probability density function to find $k$. [2]

(b) Find

(i) $P(X>1)$, [2]

(ii) $P(0.5<X<1.5)$. [2]

(c) Find the cumulative distribution function $F(x)$, giving a piecewise definition for all real $x$. [3]

(d) Find $E(X)$ and $\mathrm{Var}(X)$. [4]

(e) Find the median of $X$. [2]