19N.2.SL.TZ0.S_7

pestleMathematicsAASLPaper 219N· sl-2-4-key-features-of-graphs-intersections-using-technologysource ↗

The following table shows the probability distribution of a discrete random variable $X$ , where $a \geqslant 0$ and $b \geqslant 0$ .

Show that $b = 0.3 - a$ .

[1]

Find the difference between the greatest possible expected value and the least possible expected value.

[6]

Markscheme / solution

correct approach A1

eg $0.2 + 0.5 + b + a = 1$ , $0.7 + a + b = 1$

$b = 0.3 - a$ AG N0

[1 mark]

correct substitution into ${\text{E}}\left( X \right)$ (A1)

eg $0.2 + 4 \times 0.5 + a \times b + \left( {a + b - 0.5} \right) \times a$ , $0.2 + 2 + a \times b - 0.2a$

valid attempt to express ${\text{E}}\left( X \right)$ in one variable M1

eg $0.2 + 4 \times 0.5 + a \times \left( {0.3 - a} \right) + \left( { - 0.2} \right) \times a$ , $2.2 + 0.1a - {a^2}$ ,

$0.2 + 4 \times 0.5 + \left( {0.3 - b} \right) \times b + \left( { - 0.2} \right) \times \left( {0.3 - b} \right)$ , $2.14 + 0.5b - {b^2}$

correct value of greatest ${\text{E}}\left( X \right)$ (A1)

$2.2025$ (exact)

valid attempt to find least value (M1)

eg graph with minimum indicated, ${\text{E}}\left( 0 \right)$ and ${\text{E}}\left( {0.3} \right)$

$\left( {0{\text{, }}2.2} \right)$ and $\left( {0.3{\text{, }}2.14} \right)$ if ${\text{E}}\left( X \right)$ in terms of $a$

$\left( {0{\text{, }}2.14} \right)$ and $\left( {0.3{\text{, }}2.2} \right)$ if ${\text{E}}\left( X \right)$ in terms of $b$

correct value of least ${\text{E}}\left( X \right)$ (A1)

eg $2.14$ (exact)

difference $= 0.0625$ (exact) A1 N2

[6 marks]

Examiners’ report

[N/A]