Find an integral by choosing a suitable substitution.

Find the integral of an improper fraction.

Questions which rely on knowledge of standard integrals.

Determine the optimal frequency and size of orders given information about demand and prices.

Apply the Kruskal-Wallis test on some data to determine if a measurement differs between groups.

Implicit differentiation.

Given $x^2+y^2+ax+by=c$ find $\displaystyle \frac{dy}{dx}$ in terms of $x$ and $y$.

Factorise $x^2+bx+c$ into 2 distinct linear factors and then find $\displaystyle \int \frac{a}{x^2+bx+c }\;dx$ using partial fractions or otherwise.

Factorise $x^2+cx+d$ into 2 distinct linear factors and then find $\displaystyle \int \frac{ax+b}{x^2+cx+d}\;dx,\;a \neq 0$ using partial fractions or otherwise.

Find $\displaystyle\int \frac{ax+b}{(x+c)(x+d)}\;dx,\;a\neq 0,\;c \neq d $.

Given that $\displaystyle \int x({ax+b)^{m}} dx=\frac{1}{A}(ax+b)^{m+1}g(x)+C$ for a given integer $A$ and polynomial $g(x)$, find $g(x)$.

Find $\displaystyle \int x\sin(cx+d)\;dx,\;\;\int x\cos(cx+d)\;dx $ and hence $\displaystyle \int ax\sin(cx+d)+bx\cos(cx+d)\;dx$

Integrating by parts.

Find $ \int ax\sin(bx+c)\;dx$ or $\int ax e^{bx+c}\;dx$ 

Three parts. A sample of size $n$ is taken from $N$ where $k$ of the items are known to be defective and the task is to find the probability that more than $m$ defectives are in the sample. First part is sampling with replacement (binomial), second is sampling without replacement, (hypergeometric) and the last part uses the Poisson approximation to the first part.

Find $\frac{\mathrm{d}y}{\mathrm{d}x}$ by differentiating an implicit equation.

Integrate various functions by rewriting them as partial fractions.

Integrate the product of two functions by the method of integration by parts.

$f(x,y)$ is the PDF of a bivariate distribution $(X,Y)$ on a given rectangular region in $\mathbb{R}^2$.  Write down the limits of the integrations needed to find $P(X \ge a)$, the marginal distributions $f_X(x),\;f_Y(y)$ and the conditional probability $P(Y \le b|X \ge c)$

Calculating simple probabilities using the exponential distribution.

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

$W \sim \operatorname{Geometric}(p)$. Find $P(W=a)$, $P(b \le W \le c)$, $E[W]$, $\operatorname{Var}(W)$.

$Y \sim \operatorname{Poisson}(p)$. Find $P(Y=a)$, $P(Y \gt b)$, $E[Y],\;\operatorname{Var}(Y)$.

Given a normal distribution $X \sim N(m,\sigma^2)$ find $P(X \lt a),\; a \lt m$ and the conditional probability $P(X \gt b | X \gt c)$ where $b \lt m$ and $c \gt m$.

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

3 Repeated integrals of the form $\int_a^b\;dx\;\int_c^{f(x)}g(x,y)\;dy$ where $g(x,y)$ is a polynomial in $x,\;y$ and $f(x)$ is a degree 0, 1 or 2 polynomial in $x$.

Repeated integral of the form: $\displaystyle I=\int_0^1\;dx\;\int_0^{x^{m-1}}mf(x^m+a)dy$

Given the PDF for $Y \sim \operatorname{Exp}(\lambda)$ find the CDF, $P(a \le Y \le b)$ and $\operatorname{E}[Y],\;\operatorname{Var}(Y)$

Given the parameters of a bivariate Normal distribution $(X,Y)$ find the parameters of the Normal Distributions: $aX,\;bY,\;cX+dY,\; Y|(X=f),\;X|(Y=g)$

Determine if the following describes a probability mass function.

$P(X=x) = \frac{ax+b}{c},\;\;x \in S=\{n_1,\;n_2,\;n_3,\;n_4\}\subset \mathbb{R}$.

Questions testing understanding of numerators and denominators of numerical fractions, and reduction to lowest terms.