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HL Paper 1

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 $c$ .

Chloe and Selena play a game where each have four cards showing capital letters A, B, C and D.
Chloe lays her cards face up on the table in order A, B, C, D as shown in the following diagram.

N17/5/MATHL/HP1/ENG/TZ0/10

Selena shuffles her cards and lays them face down on the table. She then turns them over one by one to see if her card matches with Chloe’s card directly above.
Chloe wins if no matches occur; otherwise Selena wins.

Chloe and Selena repeat their game so that they play a total of 50 times.
Suppose the discrete random variable X represents the number of times Chloe wins.

Show that the probability that Chloe wins the game is $\frac{3}{8}$ .

[6]

Determine the mean of X.

[3]

b.i.

Determine the variance of X.

[2]

b.ii.

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

$f\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {k\left( {\pi - {\text{arcsin}}\,x} \right)}&{0 \leqslant x \leqslant 1} \\ 0&{{\text{otherwise}}} \end{array}} \right.,\,\,{\text{where }}k{\text{ is a positive constant}}{\text{.}}$

Given that $y = \left( {\frac{{{x^2}}}{2}} \right){\text{arcsin}}\,x - \left( {\frac{1}{4}} \right){\text{arcsin}}\,x + \left( {\frac{x}{4}} \right)\sqrt {1 - {x^2}}$ , show that

State the mode of $X$ .

[1]

Find $\int {{\text{arcsin}}\,x\,{\text{d}}x}$ .

[3]

b.i.

Hence show that $k = \frac{2}{{2 + \pi }}$ .

[3]

b.ii.

$\frac{{{\text{d}}y}}{{{\text{d}}x}} = x\,{\text{arcsin}}\,x$ .

[4]

c.i.

${\text{E}}\left( X \right) = \frac{{3\pi }}{{4\left( {\pi + 2} \right)}}$ .

[5]

c.ii.

The continuous random variable X has a probability density function given by

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

Find the value of $k$ .

[4]

By considering the graph of f write down the mean of $X$ ;

[1]

b.i.

By considering the graph of f write down the median of $X$ ;

[1]

b.ii.

By considering the graph of f write down the mode of $X$ .

[1]

b.iii.

Show that $P(0 \leqslant X \leqslant 2) = \frac{1}{4}$ .

[4]

c.i.

Hence state the interquartile range of $X$ .

[2]

c.ii.

Calculate $P(X \leqslant 4|X \geqslant 3)$ .

[2]

The probability distribution of a discrete random variable, $X$ , is given by the following table, where $N$ and $p$ are constants.

Find the value of $p$ .

[2]

Given that ${\text{E}}\left( X \right) = 10$ , find the value of $N$ .

[3]

Consider two events $A$ and $A$ defined in the same sample space.

Given that ${\text{P}}(A \cup B) = \frac{4}{9},{\text{ P}}(B|A) = \frac{1}{3}$ and ${\text{P}}(B|A') = \frac{1}{6}$ ,

Show that ${\text{P}}(A \cup B) = {\text{P}}(A) + {\text{P}}(A' \cap B)$ .

[3]

(i) show that ${\text{P}}(A) = \frac{1}{3}$ ;

(ii) hence find ${\text{P}}(B)$ .

[6]

The faces of a fair six-sided die are numbered 1, 2, 2, 4, 4, 6. Let $X$ be the discrete random variable that models the score obtained when this die is rolled.

Complete the probability distribution table for $X$ .

N16/5/MATHL/HP1/ENG/TZ0/02.a

[2]

Find the expected value of $X$ .

[2]

The discrete random variable X has the following probability distribution, where p is a constant.

Find the value of p.

[2]

Find μ, the expected value of X.

[2]

b.i.

Find P(X > μ).

[2]

b.ii.

Consider two events, $A$ and $B$ , such that ${\text{P}}\left( A \right) = {\text{P}}\left( {A' \cap B} \right) = 0.4$ and ${\text{P}}\left( {A \cap B} \right) = 0.1$ .

By drawing a Venn diagram, or otherwise, find ${\text{P}}\left( {A \cup B} \right)$ .

[3]

Show that the events $A$ and $B$ are not independent.

[3]

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 $q$ .

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).

Let $X$ be a random variable which follows a normal distribution with mean $\mu$ . Given that ${\text{P}}\left( {X < \mu - 5} \right) = 0.2$ , find

${\text{P}}\left( {X > \mu + 5} \right)$ .

[2]

${\text{P}}\left( {X < \mu + 5\,\left| {\,X > \mu - 5} \right.} \right)$ .

[5]

The continuous random variable $X$ has probability density function

$f (x) = \{\begin{cases} \frac{k}{\sqrt{4 - 3 x^{2}}}, & 0 \leq x \leq 1 \\ 0, & otherwise . \end{cases}$

Find the value of $k$ .

[4]

Find $E (X)$ .

[4]

Two unbiased tetrahedral (four-sided) dice with faces labelled 1, 2, 3, 4 are thrown and the scores recorded. Let the random variable T be the maximum of these two scores.

The probability distribution of T is given in the following table.

Find the value of a and the value of b.

[3]

Find the expected value of T.

[2]