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Integration by parts
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Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Find $\displaystyle \int x\sin(cx+d)\;dx,\;\;\int x\cos(cx+d)\;dx$ and hence $\displaystyle \int ax\sin(cx+d)+bx\cos(cx+d)\;dx$
Integration By Parts
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Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Integrating by parts.

Find $\int ax\sin(bx+c)\;dx$ or $\int ax e^{bx+c}\;dx$
Apply Friedman test
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Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Friedman test, 5 subjects, 4 treatments.
How to enter algebraic expressions - Getting Started
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Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Inputting algebraic expressions into Numbas.
How to enter algebraic fractions - Getting Started
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Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Inputting ratios of algebraic expressions.
How to enter functions - Getting Started
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Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics and 1 other

Dealing with functions in Numbas.
How to enter numbers - Getting Started
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Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics and 1 other

Details on inputting numbers into Numbas.
How to enter numbers and algebraic symbols - Getting Started
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Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics and 1 other

Entering numbers and algebraic symbols in Numbas.
How to enter powers - Getting Started
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Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Information on inputting powers
Sampling with and without replaement - binomial, hypergeometric, and Poisson approximation,
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Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

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.
Use piecewise CDF to find probabilities at given points, and the expectation.
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Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics and 1 other

Given a piecewise CDF $F_X(b)$ which is discontinuous at several points, find the probabilities at those points and also find the value of $F_X(b)$ at a continuous point and the expectation.

This cdf is a step function and is therefore the cdf of a discrete random variable. This should be stated somewhere in the statement or the solution. Apart from this the question is correct.
Find indefinite integrals of hyperbolic functions
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Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics and 1 other

Find $\displaystyle \int\cosh(ax+b)\;dx,\;\;\int x\sinh(cx+d)\;dx$ .

Advice tidied up.
Find mean and standard deviation of differences between samples
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Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics and 1 other

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 $y_i-x_i,\;i=1,\dots 5$ .
Calculate probabilities using the exponential distribution,
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Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Calculating simple probabilities using the exponential distribution.
Construct PDF and find CDF,
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Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

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 $P(X \lt a)$ .
Double integral - limit is a polynomial,
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Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

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 $x$ .
Double integral,
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Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Repeated integral of the form: $\displaystyle I=\int_0^1\;dx\;\int_0^{x^{m-1}}mf(x^m+a)dy$
Is the given function a probability mass function?, , ,
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Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Determine if the following describes a probability mass function.

$P(X=x) = \frac{ax+b}{c},\;\;x \in S=\{n_1,\;n_2,\;n_3,\;n_4\}\subset \mathbb{R}$ .
Probability of not choosing any from a subset
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Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Given subset $T \subset S$ of $m$ objects in $n$ find the probability of choosing without replacement $r\lt n-m$ from $S$ and not choosing any element in $T$ .
Calculate probability, CDF, expected value and variance of binomial distribution,
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Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

$X \sim \operatorname{Binomial}(n,p)$ . Find $P(X=a)$ , $P(X \leq b)$ , $E[X],\;\operatorname{Var}(X)$ .
Calculate probability, expected value and variance of a geometric distribution,
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Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

$W \sim \operatorname{Geometric}(p)$ . Find $P(W=a)$ , $P(b \le W \le c)$ , $E[W]$ , $\operatorname{Var}(W)$ .
Calculate probability, expected value and variance of a Poisson distribution,
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Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

$Y \sim \operatorname{Poisson}(p)$ . Find $P(Y=a)$ , $P(Y \gt b)$ , $E[Y],\;\operatorname{Var}(Y)$ .
Bivariate continuous distribution - marginals and conditional probability,
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Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

$f(x,y)$ 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 $P(Y \le b|X \ge c)$
Find CDF of given exponential distribution, and expectation and variance,
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Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Given the PDF for $Y \sim \operatorname{Exp}(\lambda)$ find the CDF, $P(a \le Y \le b)$ and $\operatorname{E}[Y],\;\operatorname{Var}(Y)$
PDF and CDF of continuous uniform random variable,
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Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

$X$ is a continuous uniform random variable defined on $[a,\;b]$ . Find the PDF and CDF of $X$ and find $P(X \ge c)$ .
Conditional probability in normal distribution,
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Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

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 $c \gt m$ .
Find parameters of normal distributions from bivariate distribution,
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Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Given the parameters of a bivariate Normal distribution $(X,Y)$ find the parameters of the Normal Distributions: $aX,\;bY,\;cX+dY,\; Y|(X=f),\;X|(Y=g)$
Reduce fractions to lowest terms
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Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Questions testing understanding of numerators and denominators of numerical fractions, and reduction to lowest terms.
Solve a linear equation
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Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Solve for $x$ : $\displaystyle ax+b = cx+d$
Solve an equation in algebraic fractions
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Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Solve for $x$ : $\displaystyle \frac{px+s}{ax+b} = \frac{qx+t}{cx+d}$ with $pc=qa$ .

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