$A,\;B$ $2 \times 2$ matrices. Find eigenvalues and eigenvectors of both. Hence or otherwise, find $B^n$ for largish $n$.

Rolling a pair of dice. Find probability that at least one die shows a given number.

Putting a pair of linear equations into matrix notation and then solving by finding the inverse of the coefficient matrix. 

An experiment is performed twice, each with $5$ outcomes

$x_i,\;y_i,\;i=1,\dots 5$ . Find mean and s.d. of their differences $y_i-x_i,\;i=1,\dots 5$.

Given a probability mass function $P(X=i)$ with outcomes $i \in \{0,1,2,\ldots 8\}$, find the expectation $E$ and $P(X \gt E)$.

Multiple response question (3 correct out of 6) re properties of convergent and divergent sequences. Selection of questions from a pool.

Multiple response question (4 correct out of 8) covering properties of convergent and divergent sequences and boundedness of sets. Selection of questions from a pool.

Given  $P(A)$, $P(A\cup B)$, $P(B^c)$ find $P(A \cap B)$, $P(A^c \cap B^c)$, $P(A^c \cup B^c)$ etc..

$X \sim \operatorname{Binomial}(n,p)$. Find $P(X=a)$, $P(X \leq b)$, $E[X],\;\operatorname{Var}(X)$.

rebelmaths

(Green’s theorem). $\Gamma$ a rectangle, find: $\displaystyle \oint_{\Gamma} \left(ax^2-by \right)\;dx+\left(cy^2+px\right)\;dy$.

Find $\displaystyle \int_{\Gamma} \left(x+y \right)\;dx+\left(y-x\right)\;dy,\;\Gamma$ is the line from $(0,0)$ to $(a,b)$.

Double integrals (2) with numerical limits

$x_n=\frac{an^2+b}{cn^2+d}$. Find the least integer $N$ such that $\left|x_n -\frac{a}{c}\right| < 10 ^{-r},\;n\geq N$, $2\leq r \leq 6$. Determine whether the sequence is increasing, decreasing or neither.

Four questions on finding least upper bounds and greatest lower bounds of various sets.

$x_n=n^k t^n$ where $k$ is a positive integer and $t$ a real number with $0 < t<1$. Find the smallest integer $N$ such that $(m+1)^k t^{m+1} \leq m^k t^m$ for all $m \geq N$. 

Arrivals given by exponential distribution, parameter $\theta$ and $Y$, sample mean on inter-arrival times. Find and calculate unbiased estimator for $\theta$.

Given a PDF $f(x)$ on the real line with unknown parameter $t$ and three random observations, find log-likelihood and MLE $\hat{t}$ for $t$. 

Using a random sample from a population with given mean and variance, find the expectation and variance of three estimators of $\mu$. Unbiased, efficient?

Given sum of sample from a Normal distribution with unknown mean $\mu$ and known variance $\sigma^2$. Find MLE of $\mu$ and one of four functions of $\mu$.

Two numbers are drawn at random without replacement from the numbers m to n.

Find the probability that both are odd given their sum is even.

Two ordered data sets, each with 10 numbers. Find the sample standard deviation for each and for their sum.

Given 32 datapoints in a table find their minimum, lower quartile, median, upper quartile, and maximum.

Two ordered data sets, each with 10 numbers. Find the sample standard deviation for each and for their sum.

The random variable $X$ has a PDF which involves a parameter $c$. Find the value of $c$. Find the distribution function $F_X(x)$ and $P(a \lt X \lt b)$.

Also find the expectation $\displaystyle \operatorname{E}[X]=\int_{-\infty}^{\infty}xf_X(x)\;dx$.

Very elementary matrix multiplication. 

Multiplication of $2 \times 2$ matrices.

Find the determinant and inverse of three $2 \times 2$ invertible matrices.

Sample of size $24$ is given in a table. Find sample mean, sample standard deviation, sample median and the interquartile range.

A box contains $n$ balls, $m$ of these are red the rest white.

$r$ are drawn without replacement.

What is the probability that at least one of the $r$ is red?

$A,\;B$ $2 \times 2$ matrices. Find eigenvalues and eigenvectors of both. Hence or otherwise, find $B^n$ for largish $n$.