DP Mathematics: Analysis and Approaches Questionbank

A discrete random variable $X$ has the probability distribution given by the following table.

Given that $\text{E}\left(X\right)=\frac{19}{12}$, determine the value of $p$ and the value of $q$.

Show that $\sqrt{16+k}-\sqrt{k}=\sqrt{k}\sqrt{16+k}$.

Show that $\text{Var}\left(X\right)=\frac{n+1}{{\left(n+2\right)}^2\left(n+3\right)}$.

Show that $\text{E}\left(X\right)=\frac{n+1}{n+2}$.

Determine the value of $m$.

Given that $P\left(\left|X-m\right|\le a\right)=0.3$, determine the value of $a$.

Find the value of $k$.

Find $\text{E}\left(X\right)$.

A continuous random variable $X$ has the probability density function

$f\left(x\right)=\left\{\begin{array}{cc}\frac{2}{\left(b-a\right)\left(c-a\right)}\left(x-a\right),& a\le x\le c\\ \frac{2}{\left(b-a\right)\left(b-c\right)}\left(b-x\right),& c<x\le b\\ 0,& \text{otherwise}\end{array}\right.$.

The following diagram shows the graph of $y=f\left(x\right)$ for $a\le x\le b$.

Given that $c\ge \frac{a+b}{2}$, find an expression for the median of $X$ in terms of $a, b$ and $c$.

A continuous random variable X has the probability density function $f$ given by

.

Find P(0 ≤ X ≤ 3).

Find E(T).

Given that Var(X) = 0.8419, find Var(T).

Find the value of a and the value of b.

Find the expected value of T.

Show that $a=\frac{2}{3}$.

Find .

Given that , and that 0.25 < s < 0.4 , find the value of s.

Find the value of $k$.

By considering the graph of f write down the mean of $X$;

By considering the graph of f write down the median of $X$;

By considering the graph of f write down the mode of $X$.

Show that $P(0⩽X⩽2)=\frac{1}{4}$.

Hence state the interquartile range of $X$.

Calculate $P(X⩽4|X⩾3)$.

Show that $a=32$ and $b=\frac{1}{12}$.

Find $\text{E}(X)$.

Find $\text{Var}(X)$.

Find the median of $X$.

Find $\text{E}(Y)$.

Find $\text{P}(Y⩾3)$.

Find the probability that on a randomly selected day, Steffi does not visit Will’s house.

Copy and complete the probability distribution table for Y.

Hence find the expected number of times per day that Steffi is fed at Will’s house.

In any given year of 365 days, the probability that Steffi does not visit Will for at most $n$ days in total is 0.5 (to one decimal place). Find the value of $n$.

Show that the expected number of occasions per year on which Steffi visits Will’s house and is not fed is at least 30.

Complete the probability distribution table for $X$.

Find the expected value of $X$.

Determine the value of $\text{E}(X^2)$.

Find the value of $\text{Var}(X)$.

Find the value of $p$.

Given that $\text{E}\left(X\right)=10$, find the value of $N$.

$\frac{\text{d}y}{\text{d}x}=x\text{arcsin}x$.

State the mode of $X$.

$\text{E}\left(X\right)=\frac{3\pi }{4\left(\pi +2\right)}$.

Find $\int \text{arcsin}x\text{d}x$.

Hence show that $k=\frac{2}{2+\pi }$.

Find $q$.