20N.2.SL.TZ0.S_3

pestleMathematicsAISLPaper 220N· sl-4-7-discrete-random-variablessource ↗

A discrete random variable $X$ has the following probability distribution.

Find an expression for $q$ in terms of $p$ .

[2]

Find the value of $p$ which gives the largest value of $E (X)$ .

[3]

b.i.

Hence, find the largest value of $E (X)$ .

[1]

b.ii.

Markscheme / solution

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

evidence of summing probabilities to $1$ (M1)

eg $q + 4 p^{2} + p + 0.7 - 4 p^{2} = 1, 1 - 4 p^{2} - p - 0.7 + 4 p^{2}$

$q = 0.3 - p$ A1 N2

[2 marks]

correct substitution into $E (X)$ formula (A1)

eg $0 \times (0.3 - p) + 1 \times 4 p^{2} + 2 \times p + 3 \times (0.7 - 4 p^{2})$

valid approach to find when $E (X)$ is a maximum (M1)

eg max on sketch of $E (X)$ , $8 p + 2 + 3 \times (- 8 p) = 0$ , $\frac{- b}{2 a} = \frac{- 2}{2 \times (- 8)}$

$p = \frac{1}{8} (= 0.125)$ (exact) (accept $x = \frac{1}{8}$ ) A1 N3

[3 marks]

b.i.

$2.225$

$\frac{89}{40}$ (exact), $2.23$ A1 N1

[1 mark]

b.ii.

Examiners’ report

[N/A]

b.i.

[N/A]

b.ii.