Home page for accesible maths Math230, 2017-18: Workshop, Coursework & Quiz Questions MATH230 Week 01 - Moodle-assessed problems MATH230 Week 01 - Assessed problems (coursework)

Style control - access keys in brackets

Font (2 3) - + Letter spacing (4 5) - + Word spacing (6 7) - + Line spacing (8 9) - +

MATH230 Week 01 - Workshop problems

Please have a go at the problems before the workshop so that in the workshop you can focus on those problems which you had trouble with.

W01.1 Jabberwocky

The brave knight is lost in the Tulgey Wood. Three paths lead on from where he stands at the Tumtum tree, clutching his vorpal sword. Unbeknownst to him, all three paths lead out of the wood; however, just off the left path lies the Jubjub bird, sleeping. Two times in three a knight walking this path would pass the bird without waking it, but if awoken the Jubjub bird would carry him up to its nest of hungry young. Along the right path lurks the frumious Bandersnatch against which the knight has no hope of prevailing. The middle path leads to the Jabberwock; if the knight encounters the Jabberwock then he might fall victim to its jaws that bite, its claws that snatch or its beguiling eyes of flame; else he will defeat it with a snicker-snack of his vorpal sword; each of the four possibilities is equally likely. The knight is as likely as not to forge straight ahead down the centre path; the right and left paths appeal equally to him.

(a)

What is the probability that the knight makes it out of the tulgey wood alive?
(b)

The brave knight does not emerge from the wood; what is the probability that he encountered the Jabberwock?

Hint: It may be helpful to define: ‘A’=‘Alive’, ‘D’=‘Dead’, L=‘left path’, R=‘right’ and C=‘centre’.

W01.2 Pregnancy

An ectopic pregnancy is twice as likely to develop when a pregnant woman is a smoker as it is when she is a non smoker. If 32% of women of childbearing age are smokers, what is the probability that a woman having an ectopic pregnancy is a smoker? You may find it helpful to define the following shorthand: $A\equiv\mbox{ ectopic pregnancy}$ , $B\equiv\mbox{ smoker}$ .

W01.3 Discrete cdf

A discrete random variable, $R$ , has a cdf of

\displaystyle F_{R}(r)=\left\{\begin{array}[]{ll}0&\quad r<0\\ 1/9&\quad 0\leq r<1\\ 5/9&\quad 1\leq r<2\\ 1&\quad 2\leq r\end{array}\right.

(a)

What is $\operatorname{\mathsf{P}}\left({R>1}\right)$ ?
(b)

What is $\operatorname{\mathsf{P}}\left({R\geq 1}\right)$ ?
(c)

What is the pmf of $R$ ?
(d)

What is the median of $R$ ?

W01.4 Discrete pmf to cdf

Obtain the cdf of a random variable with the following pmf.

\displaystyle p_{R}(r)=\left\{\begin{array}[]{ll}(1-\theta)^{r}\theta&\quad r=% 0,1,2,3,\dots\\ 0&\quad\mbox{otherwise.}\end{array}\right.

Hint Study Figure 2.1 in your notes. Your function should work for any real number $r\in\mathbb{R}$ . First calculate $F$ at the integers, being careful to distinguish the index of summation from the upper limit of summation. Secondly extend from the integers to the real line.

W01.5 Continuous cdf

Let the cdf of a continuous random variable $X$ be

\displaystyle F_{X}(x)=\left\{\begin{array}[]{ll}0&\quad x<10\\ (x-10)/10&\quad 10\leq x\leq 20\\ 1&\quad x>20\end{array}\right.

Find the pdf of $X$ .

W01.6 pdf

The random variable $X$ has pdf

\displaystyle f_{X}(x)=\left\{\begin{array}[]{ll}2x&\quad 0<x<1\\ 0&\quad\text{otherwise}\end{array}\right.

(a)

Find $\operatorname{\mathsf{P}}\left({X<{1/2}}\right)$ .
(b)

Find $\operatorname{\mathsf{P}}\left({X>{3/4}\mid X>{1/2}}\right)$ .

W01.7 pdfcst

The random variable $X$ has probability density function

f_{X}(x)=\left\{\begin{array}[]{ll}a(4-x^{2})&\quad-2<x<2\\ 0&\quad\mbox{otherwise.}\end{array}\right.

(i)

Find the constant, $a$ , and the cumulative distribution function of $X$ and, hence or otherwise, evaluate $\operatorname{\mathsf{P}}(-1<X<0)$ .
(ii)

Find the median, $q_{5/32}$ and $q_{27/32}$ of $X$ .