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Given a random variable $X$  normally distributed as $\operatorname{N}(m,\sigma^2)$ find probabilities $P(X \gt a),\; a \gt m;\;\;P(X \lt b),\;b \lt m$.

Number Theory. 

Given $n \in \mathbb{N}$ find $\mu(n),\;\tau(n),\;\sigma(n),\;\phi(n).$

Solving three simultaneous congruences using the Chinese Remainder Theorem:

\[\begin{eqnarray*} x\;&\equiv&\;b_1\;&\mod&\;n_1\\ x\;&\equiv&\;b_2\;&\mod&\;n_2\\x\;&\equiv&\;b_3\;&\mod&\;n_3 \end{eqnarray*} \] where $\operatorname{gcd}(n_1,n_2,n_3)=1$

Solving two simultaneous congruences:

\[\begin{eqnarray*} c_1x\;&\equiv&\;b_1\;&\mod&\;n_1\\ c_2x\;&\equiv&\;b_2\;&\mod&\;n_2\\ \end{eqnarray*} \] where $\operatorname{gcd}(c_1,n_1)=1,\;\operatorname{gcd}(c_2,n_2)=1,\;\operatorname{gcd}(n_1,n_2)=1$

Statistics and probability. Practice exam, one-way Anova for PSY2010.

Given two propositions in mathematics using quantifiers, choose the corresponding negation of the proposition. For example, the negation of: $\displaystyle \exists a \in \mathbb{R^+},\;\exists b \in \mathbb{N},\;\exists c \in \mathbb{N}\;\left[(c \lt b+1) \land \left(\frac{1}{2^n} \geq 3a\right)\right]$

Practice questions on these topics.

Find gcd g of two positive integers x, y and also find integers a, b such that ax+by=g with prescribed intervals for a and b.

One-way ANOVA example

Given $m \in \mathbb{N}$, find all $n \in \mathbb{N}$ such that $\phi(n)=m$ and enter the largest and second largest if they exist.

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Given one solution of the quadratic equation in $\mathbb{Z}_n$ where $n=pq$ is a product of two primes find the other 3 solutions.

Calculations in $\mathbb{Z_n}$ for three values of $n$.     

Questions about logical predicates, and basic set theory concepts.

Solving a pair of congruences of the form \[\begin{align}x &\equiv b_1\;\textrm{mod} \;n_1\\x &\equiv b_2\;\textrm{mod}\;n_2 \end{align}\] where $n_1,\;n_2$ are coprime.

Given $\frac{a}{b} \in \mathbb{Q}$ for suitable choices of $a$ and $b$, find all $n \in \mathbb{N}$ such that $\phi(n)=\frac{a}{b}n$.

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Given $m \in \mathbb{N}$, find the smallest natural number $n \in \mathbb{N}$ with $\tau(n)=m$ divisors.

Multiple correlation question. Given the correlation coefficent of $Y$ with $X_1$ is $r_{01}$, the correlation coefficent of $Y$ with $X_2$ is $r_{02}$ and the correlation coefficent of $X_1$ with $X_2$ is $r_{12}$ then explain the proportion of variablity of $Y$. Also find the partial corr coeff between $Y$ and $X_2$ after fitting $X_1$ and find how much of the remaining variability in $Y$ is explained by $X_2$ after fitting $X_1$.

Multiplication of $2 \times 2$ matrices.

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

Questions on matrix arithmetic.

Very elementary matrix multiplication. 

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

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

Given a 3x3 matrix with very big elements, perform row operations to find a matrix with single-digit elements. Then reduce that to an upper triangular matrix, and hence find the determinant.

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

Exercise using a given uniform distribution $Y$, calculating the expectation and variance as well as asking for the CDF. Also finding  $P( b \lt Y \lt c)$ for given values of $b,\;c$.

Statistics and probability. One question on multiple and partial correlation.