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$f(x,y)$ 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.

Express $\displaystyle b+  \frac{dx+p}{x + q}$ as an algebraic single fraction. 

Express $\displaystyle \frac{ax+b}{x + c} \pm  \frac{dx+p}{x + q}$ as an algebraic single fraction over a common denominator. 

Express $\displaystyle \frac{ax+b}{cx + d} \pm  \frac{rx+s}{px + q}$ as an algebraic single fraction over a common denominator. 

Find $B$ and $C$ such that $x^2+bx+c = (x+B)^2+C$.

Find $a$, $B$ and $C$ such that $ax^2+bx+c = a(x+B)^2+C$.

Harder questions testing addition, subtraction, multiplication and division of numerical fractions and reduction to lowest terms. They also test BIDMAS in the context of fractions.

Express the equation of the given line in the form $y=mx+c$.

Find the equation of the straight line which passes through the points $(a,b)$ and $(c,d)$.

Find the equation of the straight line perpendicular to the given line that passes through the given point $(a,b)$.

Questions testing understanding of numerators and denominators of numerical fractions.

Reducing fractions to their lowest form by cancelling common factors in the numerator and denominator. There are 4 questions. 

Questions testing understanding of the index laws.

Questions testing understanding of the index laws.

Questions testing understanding of the index laws.

A question testing the application of the Cosine Rule when given three side lengths. In this question, the triangle is always acute. A secondary application is finding the area of a triangle.

A question testing the application of the Cosine Rule when given three side lengths. In this question, the triangle is always obtuse. A secondary application is finding the area of a triangle.