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

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.

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 $\operatorname{gcd}(c_1,n_1)=1,\;\operatorname{gcd}(c_2,n_2)=1,\;\operatorname{gcd}(n_1,n_2)=1$

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 $\operatorname{gcd}(n_1,n_2,n_3)=1$

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

Questions about logical predicates, and basic set theory concepts.

Practice questions on these topics.

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

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.

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 $b,\;c$.

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

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

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

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.

Very elementary matrix multiplication. 

Provided with information on a sample with sample mean and standard deviation, but no information on the population variance, use the t test to either accept or reject a given null hypothesis.

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

Questions on matrix arithmetic.

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

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

Converting odds to probabilities.

Question on $\displaystyle{\lim_{n\to \infty} a^{1/n}=1}$. Find least integer $N$ s.t.  $\ \left |1-\left(\frac{1}{c}\right)^{b/n}\right| \le10^{-r},\;n \geq N$

Normal distribution $X \sim  N(\mu,\sigma^2)$ given. Find $P(a \lt X \lt b)$. Find expectation, variance, $P(c \lt \overline{X} \lt d)$ for sample mean $\overline{X}$.

Given a large number of gambles, find the expected profit.

Let $x_n=\frac{an+b}{cn+d},\;\;n=1,\;2\ldots$. Find  $\lim_{x \to\infty} x_n=L$ and find least $N$ such that $|x_n-L| \le 10^{-r},\;n \geq N,\;r \in \{2,\;3,\;4\}$.

Seven standard elementary limits of sequences. 

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