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

rebelmaths

Solving an equation of the form $ax \equiv b\;\textrm{mod}\;n$ where $a$ and $n$ are coprime.

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

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

Note that there are different versions of the upper and lower quartiles, so you may want to include your own versions - see the user defined functions in the question.

$x_n=\frac{an^2+b}{cn^2+d}$. Find the least integer $N$ such that $\left|x_n -\frac{a}{c}\right| < 10 ^{-r},\;n\geq N$, $2\leq r \leq 6$. Determine whether the sequence is increasing, decreasing or neither.

$x_n=n^k t^n$ where $k$ is a positive integer and $t$ a real number with $0 < t<1$. Find the smallest integer $N$ such that $(m+1)^k t^{m+1} \leq m^k t^m$ for all $m \geq N$. 

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 an equation of the form $ax \equiv\;b\;\textrm{mod}\;n$  where $\operatorname{gcd}(a,n)|r$. In this case we can find all solutions. The user is asked for the two greatest.

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

$x_n=\frac{an+b}{cn+d}$. Find the least integer $N$ such that $\left|x_n -\frac{a}{c}\right| \le 10 ^{-r},\;n\geq N$, $2\leq r \leq 6$.

$W \sim \operatorname{Geometric}(p)$. Find $P(W=a)$, $P(b \le W \le c)$, $E[W]$, $\operatorname{Var}(W)$.

Basic data structures and maths/stats functionality given.

You can configure the rest.

rebelmaths

Two numbers from a set of $5$ numbers are chosen at random, without replacement. Find the distribution $X$ of their sum and $E[X]$.