Problem

Jen has one fair $4$-sided red die and one fair $4$-sided blue die.

• The red die has sides numbered $1$, $2$, $3$ and $4$
• The blue die has sides numbered $1$, $3$, $5$ and $7$

The discrete random variable $R$ represents the score from one roll of the red die. The discrete random variable $B$ represents the score from one roll of the blue die.

(a) Write down the name of the distribution of $R$

(b) Find $P(R<3)$

(c) Write down the value of

(i) $E(R)$

(ii) $E(B)$

(d) Showing your working, find $\operatorname{Var}(B)$

Jen rolls each die once.

(e) Find $P(R+B\leqslant5)$

(f) Find $P(R^2<B)$

The random variable $D$ is defined as the magnitude of the difference between the score on the red die and the score on the blue die. The table below shows the cumulative distribution function of $D$

$d$	$0$	$1$	$2$	$3$	$4$	$5$	$6$
$F(d)$	$\dfrac18$	$\dfrac38$	$\dfrac9{16}$	$\dfrac34$	$p$	$\dfrac{15}{16}$	$1$

(g) Showing your working, find the value of $p$