Quiz

What does the following statement mean?
$x~\in~A~$

$x$ is an element of the set A

$x$ is a subset of the set A

$x$ is a proper subset of the set A

the intersection of $x$ and A

none of the above

What does the following statement mean?
$\forall~x$

the element $x$

there exists a value of $x$

for all values of $x$

implies $x$

the number of elements in $x$

the empty set $x$

none of the above

What does the following set representation mean?
$\mathbb{Z}$

the real numbers

the natural numbers {1,2,3,...}

the integers {..,-2,-1,0,1,2…}

the rational numbers

the complex numbers

none of the above

Which of the following set identities describe the below interval representation?
$(a,b]$

$\{x~\in~\mathbb{R};~a~\leq~x~\leq~b\}$

$\{x~\in~\mathbb{R};~a~<~x~\leq~b\}$

$\{x~\in~\mathbb{R};~a~\leq~x~<~b\}$

$\{x~\in~\mathbb{R};~a~<~x~<~b\}$

none of the above

For the set $A$ defined below, what is $|A|$ ?
$A=\{2,3,4,7,11\}$

none of the above

What does the following symbol represent?
$\Omega$

a random experiment

an outcome

a sample space

an event

a probability

none of the above

What do we call a subset of the sample space?

a random experiment

an outcome

a sample element

an event

a probability

none of the above

Which of the following describe two sets which are mutually exclusive?

$A~\cap~B~=~\emptyset~$

$A~\cup~B~=~\Omega$

$A~\cap~B^c~=~A$

$A^c~\cap~B~=~B$

$A^c~\cup~B^c~=~\Omega$

$A^c~\cap~B^c~=~\emptyset$

What is the below statement called?
$P(A~\cup~B)~=~P(A)+P(B)~-~P(A~\cap~B)$

Law of the unconcious statistician

Law of intersection

Law of union

Bayes Theorem

None of the above

Which of the following best describes a random variable?

random

variable

function

set

all of the above

none of the above

For a random variable $X$ , what is its domain?

$\mathbb{R}$

$X$

$\Omega$

$\mathbb{Z}$

$[0,1]$

Which of the following sets are discrete?

$\mathbb{R}$

$\mathbb{Z}$

$(0,1)$

$[0,1]$

${0,1}$

$\mathbb{Q}$

None of the above

All of the above

Pick all conditions necessary for a probability mass function (pmf) $p(x)$

$p(x)~>~0$

$p(x)\geq~0$

$p(x)~<~1$

$p(x)~\leq~1$

$\Sigma_{x}~p(x)~=~1$

$\Sigma_{x}~x~p(x)~=~1$

none of the above

What is the domain of a probability mass function $p(x)$ ?

$\Omega$

$X$

$Range(X)$

$\mathbb{Z}$

$\mathbb{R}$

$\mathbb{Z}^+$

$[0,1]$

None of the above

What is the codomain of a probability mass function $p(x)$ ?

$\Omega$

$X$

$Range(X)$

$\mathbb{Z}$

$\mathbb{Z}^+$

$[0,1]$

None of the above

Which of the following defines a cumulative distribution function (cdf) $F(x)$ ?

$P(X~\leq~x)$

$P(X~\in~(-\infty,~x])$

$P(-\infty~<~X~\leq~x)$

all of the above

none of the above

Given the following definition of the pdf $f(x)$ , what is k?

$f(x)~=~\begin{cases}~kx,~x~\in~[0,1]~\\~0,~otherwise~\end{cases}~$

For $F(x)$ defined below, what is $P(X>1)$ ? Please give your answer to 3 decimal places.

$F(x)~=~1-\exp(-2x)$

For $E[X]~=3$ and $E[Y]~=~2$ , what is $E[2X+5Y]$ ?

For $Var[X]~=~2$ , what is $Var[2X+3]$ ?

For the following pmf, what is $P(X<21~|~X~\geq~E[X])$ ?

$P(X=x)~=~\begin{cases}~0.4,~x~=~7~\\~0.35,~x~=~13~\\~0.1,~x~=~17\\~0.15,~x~=~21\\~0,~\text{otherwise}~\end{cases}$

For the previous pmf, what is the mode?

If $X$ is Binomial( $n,p$ ), what is $E[X]$ ?

np(1-p)

none of the above

For the following pmf, if $E[X]~=~1.2$ and $Var[X]~=~0.56$ , what is the value of a?

$P(X=x)~=~\begin{cases}~a,~x=0\\~b,~x~=~1~\\~c,~x~=~2\\~0,~\text{otherwise}~\end{cases}~$

In the previous question, what is b?

Which of the following describe the Normal distribution?

symmetric

unimodal

continuous

Gaussian

all of the above

none of the above

What is the variance of a Poisson( $\lambda$ ) distribution:
$p(x)~=~\frac{\lambda^x}{x!}e^{-\lambda}$

$\lambda$

$\lambda^2$

$\frac{1}{\lambda}$

$\frac{1}{\lambda^2}$

None of the above

What is the expectation of an Exponential $(\lambda)$ distribution?

$\lambda$

$\lambda^2$

$\frac{1}{\lambda}$

$\frac{1}{\lambda^2}$

None of the above

For a Uniform(a,b) distribution, what is the expectation?
$f(x)~=~\begin{cases}~\frac{1}{b-a},~a~\leq~x~\leq~b~\\~0,~otherwise~\end{cases}$

a-b

b-a

$\frac{1}{a}-\frac{1}{b}$

$\frac{1}{b}-\frac{1}{a}$

$\frac{1}{b-a}$

$\frac{a+b}{2}$

None of the above

The time T days between rain showers in a city is modelled by an Exponential distribution with mean 1.25. What is the distribution for the number of showers in one day?

Normal $(0.8)$

Normal $(1.25)$

Exponential $(0.8)$

Exponential $(1.25)$

Poisson $(0.8)$

Poisson $(1.25)$

None of the above

Which of the following distributions are discrete?

Binomial

Normal

Uniform[a,b]

Uniform{1,2,...n}

Poisson

Exponential

All of the above

None of the above

For a Normal distribution, which of the following expressions are equivalent to $P(X>a)$ ?

$1-P(X~\leq~a)$

$P(X<-a)$

$1-[P(X\leq~a)~+~P(-a~<~X~\leq~a)]$

All of the above

None of the above

Which of the following statements are always true?

$P(A~\cup~B)~>P(B)~$

$P(A~\cap~B)~\leq~P(A)$

$P(A~\cap~B)~\leq~P(A)P(B)$

$P(A~\cup~B)~\leq~P(A)~+~P(B)$

$P(A~\vert~B)~>~P(A)$

None of the above

Which of the following are always equal to $P(A~\cap~B)$ ?

$P(A)-P(A~\slash~B)~$

$P(A^c~\cap~B^c)$

$P(A)~\times~P(B)~$

$P(A~\vert~B)P(A)$

$P(A~\vert~B)\times~[P(A~\cup~B)~-~P(A~\backslash~B)]$

None of the above

The law of total probability states, for some event B and set of events $A_1,~A_2,~\dots,~A_n$ :
$P(B)~=~\sum_{i=1}^{n}~P(B~\vert~A_i)P(A_i)$
Which one of the following conditions is necessary and sufficient?

$A_1,~A_2,~\dots~A_n$ are independent

$A_1,~A_2,~\dots~A_n$ are mutually exclusive

$A_1,~A_2,~\dots~A_n$ are exhaustive i.e. their union covers the entire sample space

$A_1,~A_2,~\dots~A_n$ are a partition of the sample space

None of the above

Suppose that a test for a rare disease has a sensitivity (probability of positive test given the disease is present) of 0.9. Let D represent the disease being present, and T represent a positive test. Which of the following statements is true?

$P(D~\vert~T)~=~0.9$

$P(T~\vert~D)~=~0.9$

$P(T~\cap~P)~=~0.9~$

$P(T)~=~0.9$

$P(D)~=~0.9$

Suppose again that a test for a rare disease has a sensitivity (probability of positive test given the disease is present) of 0.9 and a specificity (probability of negative test given the disease is absent) of 0.8. Suppose that, conditional on a positive test, the chance of having the disease is 0.5. Find the prevalence of the disease in the population, giving your answer to 2 decimal places. You may wish to draw a tree diagram first.

A student is planning a weekend activity, and decides what to either go to the park or to the cinema, depending on the weather. The student’s plan is shown in the tree diagram found at the following link:

https://warwick.ac.uk/fac/sci/statistics/currentstudents/newstudents/prep-work/probabilityrefresher/quiz/weekendactivity/

Using this diagram, calculate P(Raining).

Using the weekend acitivty tree diagram, calculate P(Cinema | Raining).

Using the weekend acitivty tree diagram, calculate P(Sunny $\cap$ Park).

Using the weekend acitivty tree diagram, calculate P(Park)

Using the weekend acitivty tree diagram, calculate P(Sunny | Park), giving your answer correct to two decimal places

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