For a sample of size n from a normal distribution and given the mean of the sample and the standard deviation of the distribution, find the MLE for the mean. Also the expected information and a confidence interval for the mean.

For a sample of size n from a Poisson distribution $\operatorname{Poisson}(\lambda)$ and given the mean of the sample, find the MLE for $\lambda$. Also find the expected information and a confidence interval for $\lambda$.

For a sample of size n from an Exponential distribution $\operatorname{Exponential}(\lambda)$  and given the mean of the sample, find the MLE for $\lambda$. Also find the expected information and a confidence interval for $\lambda$.

For a sample of size n from a normal distribution, given mean of the sample mean and the standard deviation , find the t-statistic corresponding to a null hypothesis $\mu=m$ and a given confidence level. Check if the result is significant at this level.

Given a linear combination $Y$ of three independent random variables with given means and variances, find the mean of variance of $Y$.

Given a discrete random variable $X$ find the expectation of $1/X$ and $e^X$.

Given 3 observations from a $\operatorname{Poisson}(\mu)$ distribution find the likelihood, the log likelihood and the MLE for $\mu$.

Given sum of sample from a Normal distribution with unknown mean $\mu$ and known variance $\sigma^2$. Find MLE of $\mu$ and one of four functions of $\mu$.

Using a random sample from a population with given mean and variance, find the expectation and variance of three estimators of $\mu$. Unbiased, efficient?

Given a PDF $f(x)$ on the real line with unknown parameter $t$ and three random observations, find log-likelihood and MLE $\hat{t}$ for $t$. 

Arrivals given by exponential distribution, parameter $\theta$ and $Y$, sample mean on inter-arrival times. Find and calculate unbiased estimator for $\theta$.

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

Simple probability question. Counting number of occurences of an event in a sample space with given size and finding the probability of the event.

Given a table of the number of days in which sales were between £x1000 and £(x+1)1000 find the relative percentage frequencies of these volume of sales.

Given random set of data (between 13 and 23 numbers all less than 100), find their stem-and-leaf plot.

Find a regression equation.

Given sample data find mean, standard deviation, median, interquartile range,

Finding probabilities from a survey giving a table of data on the alcohol consumption of males. This can be easily adapted to data from other types of surveys.

Eight questions on finding least upper bounds and greatest lower bounds of various sets.

Find a regression equation.

Given data on probabilities of three levels of success of three options and projections of the profits that the options will accrue depending on the level of success, find the expected monetary value (EMV) for each option and choose the one with the greatest EMV.

Find out whether the data presented in this question follows a Poisson distribution. Uses the $\chi^2$ test.

One question on determining whether statements are propositions.

Four questions on find truth tables for various logical expressions.

Statistics and probability. 2 questions. Both simple regression. First with 8 data points, second with 10. Find $a$ and $b$ such that $Y=a+bX$. Then find the residual value for one of the data points.

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.

Number Theory. 

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

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

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.

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.

Given $m \in \mathbb{N}$, find values of $n\in \mathbb{N}$ such that $\sigma(n)=m$.

There are at most two such solutions in this question.