Home page for accesible maths Math230, 2017-18: Workshop, Coursework & Quiz Questions Coursework and Quiz marks MATH230 Week 01 - Workshop problems

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MATH230 Week 01 - Moodle-assessed problems

QZ k is due 11:59pm on the Sunday of Week k, except for the revision quiz (k=1), which is due on the Sunday ending Week 2. Numerical answers should be entered as decimals and accurate to 3dp.

Revision Quiz

M01.1 Basic probability

Nigel has $3$ cards; the first has a triangle on one side, the second has a square and the third has a circle. The other side of each card is blank. Showing you only the blank sides, Nigel asks you to choose a card at random. If you choose the triangle he rolls an unbiased four-sided die (i.e. with the numbers $1$ – $4$ on the faces, with each face equally likely to be chosen); if you choose the square he rolls a six-sided die and if you choose the circle he rolls an eight-sided die. The end result of the experiment is a shape and a number. However, Nigel does not tell you which shape you chose and does not let you see him roll the die, he says he will simply call out the final number after a dramatic pause.

(i)

What is the probability that you choose a triangle and the final number is a $3$ ?
(ii)

What is the probability that the final number is a $3$ ?
(iii)

What is the probability that the final number is not a $3$ ?
(iv)

What is the probability that the shape is a triangle or the final number is $3$ or both?
(v)

The final number is a $3$ (Nigel has just called it out); what is the probability that your shape was a triangle?

M01.2 Discrete random variable

A discrete random variable, $X$ , takes a value $x\in\{1,2,3,4\}$ with a probability of $x/10$ . It cannot take any other value.

(i)

What is $\operatorname{\mathsf{P}}\left({X\geq 3}\right)$ ?
(ii)

What is $\operatorname{\mathsf{E}}\left[{X}\right]$ ?
(iii)

What is $\operatorname{\mathsf{E}}\left[{X^{2}}\right]$ ?
(iv)

What is $\operatorname{\mathsf{E}}\left[{1/X}\right]$ ?
(v)

What is ${\operatorname{\mathsf{Var}}}\left[{X}\right]$ ?

M01.3 Continuous random variable

A continuous random variable, $Y$ , has a cumulative distribution function of $F_{Y}(y)=c-1/(1+y)^{2}$ for $y>0$ , and $0$ otherwise.

(i)

What is $c$ ?
(ii)

What is $f_{Y}(3)$ (the density, evaluated at $y=3$ )?
(iii)

What is $\operatorname{\mathsf{P}}\left({1\leq Y\leq 3}\right)$ ?
(iv)

What is $\operatorname{\mathsf{E}}\left[{1+Y}\right]$ ?
(v)

What is the third quartile of $Y$ ?

M01.4 Variance, standard deviation and linearity of expectation

A random variable, $Z$ , has $\operatorname{\mathsf{E}}\left[{Z}\right]=4$ and ${\operatorname{\mathsf{Var}}}\left[{Z}\right]=9$ .

(i)

What is $\operatorname{\mathsf{E}}\left[{10-2Z}\right]$ ?
(ii)

What is $\operatorname{\mathsf{E}}\left[{Z^{2}}\right]$ ?
(iii)

What is $\operatorname{\mathsf{E}}\left[{(10-2Z)^{2}}\right]$ ?
(iv)

What is ${\operatorname{\mathsf{StdDev}}}\left[{Z}\right]$ ?
(v)

What is ${\operatorname{\mathsf{StdDev}}}\left[{10-2Z}\right]$ ?

M01.5 Colds

The number of times that an individual contracts a cold in a given year follows a Poisson distribution with the expected number per year being $5$ . A new drug has been introduced which changes the distribution to Poisson with expected number $3$ per year for $75\%$ of the population. For the other $25\%$ of the population the drug has no appreciable effect on colds. Let $N$ be the number of colds in a year and $B$ be the event that the drug is beneficial for you.

(i)
The random variable that counts the number of colds when the drug is not beneficial is represented by
1. (A)
  
  $B\operatorname{\mid}N$ ,
2. (B)
  
  $N\operatorname{\mid}B$ ,
3. (C)
  
  $N\operatorname{\mid}B^{C}$ ,
4. (D)
  
  $N,B^{C}$ ,
5. (E)
  
  $B\operatorname{\mid}N$ .
(ii)
The distribution of the random variable in (i) is
1. (A)
  
  $\operatorname{\mathsf{Poisson}}(3)$ ,
2. (B)
  
  $\operatorname{\mathsf{Poisson}}(0.25)$ ,
3. (C)
  
  $\operatorname{\mathsf{Poisson}}(0.75)$ ,
4. (D)
  
  $\operatorname{\mathsf{Poisson}}(5)$ ,
5. (E)
  
  $\operatorname{\mathsf{Poisson}}(1)$ .
(iii)
You try the drug for a year and have $2$ colds. What is the approximate probability that the drug is beneficial for you?
1. (A)
  
  $0.75$ ,
2. (B)
  
  $0.224$ ,
3. (C)
  
  $0.051$ ,
4. (D)
  
  $0.889$ ,
5. (E)
  
  $1/2$ .

M01.6 Modelling

Which of the following distributions is the most appropriate for modelling each of these situations. (A) Poisson, (B) Bernoulli, (C) Continuous Uniform, (D) Discrete Uniform, (E) Exponential, (F) Binomial, (G) Geometric.

(i)

The number of people you need to ask before you find someone who prefers crisps to chocolate.
(ii)

The number of people in the queue for tickets, excluding you, when you first arrive at the cinema.
(iii)

The time from when you join the queue until the person at the very front of the queue has been served.
(iv)

The seat number (i.e. position from left to right, rather than row) of the oldest person in the cinema.
(v)

The time during the film showing at which your mobile phone goes off (you accidentally leave it on, and it does go off).
(vi)

The number of vegetarians in your Math230 tutorial group (excluding you).
(vii)

Whether there are any vegetarians in your Math230 tutorial group (excluding you).