Probability & Statistics

14 methods · 60 questions

★ Signature method unlocked: the technique this chapter rewards most often, with 3 worked questions.

The full chapter (14 methods · 60 worked questions) comes with Platform Access. See pricing

To find the distribution of $Y = g (X)$ for a monotone (or piecewise monotone) $g$ , work through the cumulative distribution function rather than guessing a density: write $P (Y \leq y) = P (X \leq g^{- 1} (y))$ (reverse the inequality where $g$ decreases), then differentiate to get the density, which for an increasing $g$ gives $f_{X} (g^{- 1} (y)) \frac{d}{d y} g^{- 1} (y)$ .

Trigger: a new variable defined as a function of a known one ( $e^{X}$ , $1/ V$ , $X + Y$ , $X / (X + Y)$ , a ratio).

Instances:
(i) $Y = e^{X}$ : $P (Y \leq y) = P (X \leq ln y)$ , density $f (ln y) / y$ ;
(ii) reciprocal $T = s / V$ or $1/ (X + Y)$ : invert the inequality and substitute;
(iii) a sum or ratio of two uniforms found by integrating the joint density over the region ${g \leq y}$ .

Linked questions (3)

STEP 3 2009 · Q13

The point $P$ lies on the circumference of a circle of unit radius and centre $O$ . The angle, $θ$ , between $O P$ and the positive $x$ -axis is a random variable, uniformly distributed on the interval $0 ⩽ θ < 2 π$ . The cartesian coordinates of $P$ with respect to $O$ are $(X, Y)$ . Find the probability density function for $X$ , and calculate $Var (X)$ . Show that $X$ and $Y$ are uncorrelated and discuss briefly whether they are independent.
The points $P_{i}$ ( $i = 1$ , $2$ , $\dots$ , $n$ ) are chosen independently on the circumference of the circle, as in part (i), and have cartesian coordinates $(X_{i}, Y_{i})$ . The point $\overline{P}$ has coordinates $(\overline{X}, \overline{Y})$ , where $\overline{X} = \frac{1}{n} i = 1 \sum n X_{i}$ and $\overline{Y} = \frac{1}{n} i = 1 \sum n Y_{i}$ . Show that $\overline{X}$ and $\overline{Y}$ are uncorrelated. Show that, for large $n$ , $P (∣ \overline{X} ∣ ⩽ \frac{2}{n}) \approx 0.95$ .

STEP 2 2012 · Q13

In this question, you may assume that

\int_{0}^{\infty} e^{- x^{2} /2} d x = \frac{1}{2} π

.

The number of supermarkets situated in any given region can be modelled by a Poisson random variable, where the mean is

k

times the area of the given region. Find the probability that there are no supermarkets within a circle of radius

y

.

The random variable

Y

denotes the distance between a randomly chosen point in the region and the nearest supermarket. Write down

P (Y < y)

and hence show that the probability density function of

Y

2 π y k e^{- π k y^{2}}

for

y ⩾ 0

.

Find

E (Y)

and show that

Var (Y) = \frac{4 - π}{4 π k}

STEP 2 2014 · Q12

The lifetime of a fly (measured in hours) is given by the continuous random variable

T

with probability density function

f (t)

and cumulative distribution function

F (t)

. The hazard function,

h (t)

, is defined, for

F (t) < 1

, by

h (t) = \frac{f ( t )}{1 - F ( t )} .

Given that the fly lives to at least time $t$ , show that the probability of its dying within the following $δ t$ is approximately $h (t) δ t$ for small values of $δ t$ .
Find the hazard function in the case $F (t) = t / a$ for $0 < t < a$ . Sketch $f (t)$ and $h (t)$ in this case.
The random variable $T$ is distributed on the interval $t > a$ , where $a > 0$ , and its hazard function is $t^{- 1}$ . Determine the probability density function for $T$ .
Show that $h (t)$ is constant for $t > b$ and zero otherwise if and only if $f (t) = k e^{- k (t - b)}$ for $t > b$ , where $k$ is a positive constant.
The random variable $T$ is distributed on the interval $t > 0$ and its hazard function is given by $h (t) = (\frac{λ}{θ ^{λ}}) t^{λ - 1},$ where $λ$ and $θ$ are positive constants. Find the probability density function for $T$ .