17M.2.SL.TZ1.S_9

pestleMathematicsAISLPaper 217M· sl-4-6-combined-mutually-exclusive-conditional-independence-prob-diagramssource ↗

A random variable $X$ is normally distributed with mean, $\mu$ . In the following diagram, the shaded region between 9 and $\mu$ represents 30% of the distribution.

M17/5/MATME/SP2/ENG/TZ1/09

The standard deviation of $X$ is 2.1.

The random variable $Y$ is normally distributed with mean $\lambda$ and standard deviation 3.5. The events $X > 9$ and $Y > 9$ are independent, and $P\left( {(X > 9) \cap (Y > 9)} \right) = 0.4$ .

Find ${\text{P}}(X < 9)$ .

[2]

Find the value of $\mu$ .

[3]

Find $\lambda$ .

[5]

Given that $Y > 9$ , find ${\text{P}}(Y < 13)$ .

[5]

Markscheme / solution

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

valid approach (M1)

eg $\,\,\,\,\,$ ${\text{P}}(X < \mu ) = 0.5,{\text{ }}0.5 - 0.3$

${\text{P}}(X < 9) = 0.2$ (exact) A1 N2

[2 marks]

$z = - 0.841621$ (may be seen in equation) (A1)

valid attempt to set up an equation with their $z$ (M1)

eg $\,\,\,\,\,$ $- 0.842 = \frac{{\mu - X}}{\sigma },{\text{ }} - 0.842 = \frac{{X - \mu }}{\sigma },{\text{ }}z = \frac{{9 - \mu }}{{2.1}}$

10.7674

$\mu = 10.8$ A1 N3

[3 marks]

${\text{P}}(X > 9) = 0.8$ (seen anywhere) (A1)

valid approach (M1)

eg $\,\,\,\,\,$ ${\text{P}}(A) \times {\text{P}}(B)$

correct equation (A1)

eg $\,\,\,\,\,$ $0.8 \times {\text{P}}(Y > 9) = 0.4$

${\text{P}}(Y > 9) = 0.5$ A1

$\lambda = 9$ A1 N3

[5 marks]

finding ${\text{P}}(9 < Y < 13) = 0.373450$ (seen anywhere) (A2)

recognizing conditional probability (M1)

eg $\,\,\,\,\,$ ${\text{P}}(A|B),{\text{ P}}(Y < 13|Y > 9)$

correct working (A1)

eg $\,\,\,\,\,$ $\frac{{{\text{0.373}}}}{{0.5}}$

0.746901

0.747 A1 N3

[5 marks]

Examiners’ report

[N/A]