IB Questionbank

Updates to Questionbank

User interface language: English | Español

Date	May 2018	Marks available	3	Reference code	18M.2.AHL.TZ1.H_10
Level	Additional Higher Level	Paper	Paper 2	Time zone	Time zone 1
Command term	Find	Question number	H_10	Adapted from	N/A

Question

The continuous random variable X has probability density function $f$ given by

$f\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {3ax}&,&{0 \leqslant x < 0.5} \\ {a\left( {2 - x} \right)}&,&{0.5 \leqslant x < 2} \\ 0&,&{{\text{otherwise}}} \end{array}} \right.$

Show that $a = \frac{2}{3}$ .

[3]

a.

Find ${\text{P}}\left( {X < 1} \right)$ .

[3]

b.

Given that ${\text{P}}\left( {s < X < 0.8} \right) = 2 \times {\text{P}}\left( {2s < X < 0.8} \right)$ , and that 0.25 < s < 0.4 , find the value of s.

[7]

c.

Markscheme

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

$a\left[ {\int_0^{0.5} {3x\,{\text{d}}x} + \int_{0.5}^2 {\left( {2 - x} \right)} \,{\text{d}}x} \right] = 1$ M1

Note: Award the M1 for the total integral equalling 1, or equivalent.

$a\left( {\frac{3}{2}} \right) = 1$ (M1)A1

$a = \frac{2}{3}$ AG

[3 marks]

a.

EITHER

$\int_0^{0.5} {2x\,{\text{d}}x} + \frac{2}{3}\int_{0.5}^1 {\left( {2 - x} \right)} \,{\text{d}}x$ (M1)(A1)

$= \frac{2}{3}$ A1

OR

$\frac{2}{3}\int_1^2 {\left( {2 - x} \right)} \,{\text{d}}x = \frac{1}{3}$ (M1)

so ${\text{P}}\left( {X < 1} \right) = \frac{2}{3}$ (M1)A1

[3 marks]

b.

${\text{P}}\left( {s < X < 0.8} \right) = \int_s^{0.5} {2x\,{\text{d}}x} + \frac{2}{3}\int_{0.5}^{0.8} {\left( {2 - x} \right)} \,{\text{d}}x$ M1A1

$= \left[ {{x^2}} \right]_s^{0.5} + 0.27$

$0.25 - {s^2} + 0.27$ (A1)

${\text{P}}\left( {2s < X < 0.8} \right) = \frac{2}{3}\int_{2s}^{0.8} {\left( {2 - x} \right)} \,{\text{d}}x$ A1

$= \frac{2}{3}\left[ {2x - \frac{{{x^2}}}{2}} \right]_{2s}^{0.8}$

$\frac{2}{3}\left( {1.28 - \left( {4s - 2{s^2}} \right)} \right)$

equating

$0.25 - {s^2} + 0.27 = \frac{4}{3}\left( {1.28 - \left( {4s - 2{s^2}} \right)} \right)$ (A1)

attempt to solve for s (M1)

s = 0.274 A1

[7 marks]

c.

Examiners report

[N/A]

a.

[N/A]

b.

[N/A]

c.

Syllabus sections

Topic 4—Statistics and probability » AHL 4.14—Properties of discrete and continuous random variables

Show 46 related questions

17M.1.AHL.TZ1.H_10b.iii:
By considering the graph of f write down the mode of $X$ .
19M.2.AHL.TZ2.H_10b:
Copy and complete the probability distribution table for Y.
22M.1.AHL.TZ2.8:
A continuous random variable $X$ has the probability density function

$f (x) = \{\begin{matrix} \frac{2}{(b - a) (c - a)} (x - a), & a \leq x \leq c \\ \frac{2}{(b - a) (b - c)} (b - x), & c < x \leq b \\ 0, & otherwise \end{matrix}$ .

The following diagram shows the graph of $y = f (x)$ for $a \leq x \leq b$ .

Given that $c \geq \frac{a + b}{2}$ , find an expression for the median of $X$ in terms of $a, b$ and $c$ .
SPM.2.AHL.TZ0.6a:
Find E(T).
SPM.1.AHL.TZ0.7:
A continuous random variable X has the probability density function $f$ given by

$f\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {\frac{{\pi x}}{{36}}{\text{sin}}\left( {\frac{{\pi x}}{6}} \right){\text{,}}}&{0 \leqslant x \leqslant 6} \\ {0{\text{,}}}&{{\text{otherwise}}} \end{array}} \right.$ .

Find P(0 ≤ X ≤ 3).
18M.1.AHL.TZ1.H_3b:
Find the expected value of T.
17M.2.AHL.TZ2.H_10a:
Show that $a = 32$ and $b = \frac{1}{{12}}$ .
19M.1.AHL.TZ2.H_10b.i:
Find $\int {{\text{arcsin}}\,x\,{\text{d}}x}$ .
18M.2.AHL.TZ1.H_10a:
Show that $a = \frac{2}{3}$ .
17M.1.AHL.TZ1.H_10c.ii:
Hence state the interquartile range of $X$ .
17M.2.SL.TZ2.S_10a.i:
Find $q$ .
17M.2.AHL.TZ2.H_10f:
Find ${\text{P}}(Y \geqslant 3)$ .
19M.1.AHL.TZ2.H_10c.ii:
${\text{E}}\left( X \right) = \frac{{3\pi }}{{4\left( {\pi + 2} \right)}}$ .
17M.2.AHL.TZ2.H_10c:
Find ${\text{Var}}(X)$ .
16N.2.AHL.TZ0.H_1a:
Determine the value of ${\text{E}}({X^2})$ .
22M.1.AHL.TZ1.7b:
Find $E (X)$ .
19M.2.AHL.TZ2.H_10d:
In any given year of 365 days, the probability that Steffi does not visit Will for at most $n$ days in total is 0.5 (to one decimal place). Find the value of $n$ .
18M.1.AHL.TZ1.H_3a:
Find the value of a and the value of b.
17M.1.AHL.TZ1.H_10b.i:
By considering the graph of f write down the mean of $X$ ;
19M.2.AHL.TZ2.H_10c:
Hence find the expected number of times per day that Steffi is fed at Will’s house.
18M.2.AHL.TZ1.H_10c:
Given that ${\text{P}}\left( {s < X < 0.8} \right) = 2 \times {\text{P}}\left( {2s < X < 0.8} \right)$ , and that 0.25 < s < 0.4 , find the value of s.
17M.2.AHL.TZ2.H_10d:
Find the median of $X$ .
19M.1.AHL.TZ2.H_10c.i:
$\frac{{{\text{d}}y}}{{{\text{d}}x}} = x\,{\text{arcsin}}\,x$ .
19M.1.AHL.TZ2.H_10a:
State the mode of $X$ .
17M.1.AHL.TZ1.H_10c.i:
Show that $P(0 \leqslant X \leqslant 2) = \frac{1}{4}$ .
16N.1.AHL.TZ0.H_2a:
Complete the probability distribution table for $X$ .
16N.2.AHL.TZ0.H_1b:
Find the value of ${\text{Var}}(X)$ .
17M.2.AHL.TZ2.H_10b:
Find ${\text{E}}(X)$ .
17M.1.AHL.TZ1.H_10d:
Calculate $P(X \leqslant 4|X \geqslant 3)$ .
19N.1.AHL.TZ0.H_1b:
Given that ${\text{E}}\left( X \right) = 10$ , find the value of $N$ .
20N.1.AHL.TZ0.H_1:
A discrete random variable $X$ has the probability distribution given by the following table.

Given that $E (X) = \frac{19}{12}$ , determine the value of $p$ and the value of $q$ .
22M.1.AHL.TZ1.7a:
Find the value of $k$ .
SPM.2.AHL.TZ0.6b:
Given that Var(X) = 0.8419, find Var(T).
17M.2.AHL.TZ2.H_10e:
Find ${\text{E}}(Y)$ .
19M.1.AHL.TZ2.H_10b.ii:
Hence show that $k = \frac{2}{{2 + \pi }}$ .
16N.1.AHL.TZ0.H_2b:
Find the expected value of $X$ .
17M.1.AHL.TZ1.H_10a:
Find the value of $k$ .
17M.1.AHL.TZ1.H_10b.ii:
By considering the graph of f write down the median of $X$ ;
19N.1.AHL.TZ0.H_1a:
Find the value of $p$ .
19M.2.AHL.TZ2.H_10e:
Show that the expected number of occasions per year on which Steffi visits Will’s house and is not fed is at least 30.
19M.2.AHL.TZ2.H_10a:
Find the probability that on a randomly selected day, Steffi does not visit Will’s house.
21M.2.AHL.TZ1.7a:
Show that $\sqrt{16 + k} - \sqrt{k} = \sqrt{k} \sqrt{16 + k}$ .
21M.2.AHL.TZ2.6a:
Show that $E (X) = \frac{n + 1}{n + 2}$ .
21M.2.AHL.TZ2.6b:
Show that $Var (X) = \frac{n + 1}{{(n + 2)}^{2} (n + 3)}$ .
21N.2.AHL.TZ0.7a:
Determine the value of $m$ .
21N.2.AHL.TZ0.7b:
Given that $P (|X - m| \leq a) = 0.3$ , determine the value of $a$ .

Hide related questions

Topic 4—Statistics and probability