Question 1 - A-Level Maths

Exam Board	Edexcel
Module	S1 (Statistics 1)
Topic	Normal Distribution
Type	Single tail probability P(X < a) or P(X > a)

$70 \%$ of the households in a town have a CD player and $45 \%$ have both a CD player and a personal computer (PC). 18\% have neither a CD player nor a PC.
1. Illustrate this information using a Venn diagram.
2. Find the percentage of the households that do not have a PC.
3. Find the probability that a household chosen at random has a CD player or a PC but not both.
4. The random variable $X$ has the normal distribution $\mathrm { N } \left( 2,1 \cdot 7 ^ { 2 } \right)$.
5. State the standard deviation of $X$.
6. Find $\mathrm { P } ( X < 0 )$.
7. Find $\mathrm { P } ( 0 \cdot 6 < X < 3 \cdot 4 )$.
8. The discrete random variable $X$ has probability function
$$\mathrm { P } ( X = x ) = \left\{ \begin{array} { c l } c x ^ { 2 } & x = - 3 , - 2 , - 1,1,2,3
0 & \text { otherwise. } \end{array} \right.$$
Show that $c = \frac { 1 } { 28 }$.
Calculate
1. $\mathrm { E } ( X )$,
2. $\mathrm { E } \left( X ^ { 2 } \right)$.
Calculate
1. $\operatorname { Var } ( X )$,
2. $\operatorname { Var } ( 10 - 2 X )$.
  (3 marks)