Question 4 - A-Level Maths

Exam Board	AQA
Module	S2 (Statistics 2)
Year	2011
Session	January
Topic	Discrete Probability Distributions
Type	Calculate E(X) from given distribution

A red biased tetrahedral die is rolled. The number, $X$, on the face on which it lands has the probability distribution given by
$\boldsymbol { x }$ 1 2 3 4
$\mathbf { P } ( \boldsymbol { X } = \boldsymbol { x } )$ 0.2 0.1 0.4 0.3
1. Calculate $\mathrm { E } ( X )$ and $\operatorname { Var } ( X )$.
2. The red die is now rolled three times. The random variable $S$ is the sum of the three numbers obtained. Find $\mathrm { E } ( S )$ and $\operatorname { Var } ( S )$.
A blue biased tetrahedral die is rolled. The number, $Y$, on the face on which it lands has the probability distribution given by $$\mathrm { P } ( Y = y ) = \begin{cases} \frac { y } { 20 } & y = 1,2 \text { and } 3
\frac { 7 } { 10 } & y = 4 \end{cases}$$ The random variable $T$ is the value obtained when the number on the face on which it lands is multiplied by 3 . Calculate $\mathrm { E } ( T )$ and $\operatorname { Var } ( T )$.
Calculate:
1. $\mathrm { P } ( X > 1 )$;
2. $\mathrm { P } ( X + T \leqslant 9$ and $X > 1 )$;
3. $\mathrm { P } ( X + T \leqslant 9 \mid X > 1 )$.

\(\boldsymbol { x }\)	1	2	3	4
\(\mathbf { P } ( \boldsymbol { X } = \boldsymbol { x } )\)	0.2	0.1	0.4	0.3