4.14 Continuous random variables questions for IB Maths AA

Find the value of $k$.

Find the mode of $X$.

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

Show that ${\int }_{-1}^{1}f(y)\mathrm{d}y=1$.

For a piece cut by machine A, show that $\mathrm{P}(\mathrm{\mid }Y\mathrm{\mid }>c)=\frac{1}{2}(1-c{)}^2(c+2)$, for $0\le c\le 1$.

For a piece cut by machine B, find an expression for $\mathrm{P}(\mathrm{\mid }Y\mathrm{\mid }>c)$, for $0\le c\le 1$.

Find the probability that a randomly chosen piece is rejected.

Given that a piece is rejected, find the probability that it was cut by machine A.

Show that the probability that a piece was cut by machine A, given that $\mathrm{\mid }Y\mathrm{\mid }>c$, is ${\displaystyle \frac{3c+6}{3c+10}}$, for $0\le c<1$.

Find the value of $c$ for which this probability is 0.65.

Show that ${\displaystyle {\int }_0^1{f}_n(x)\mathrm{d}x=1}$ for all $n\in {\mathbb{Z}}^{+}$.

Consider the case $n=3$.

Find $\mathrm{E}(X_3)$.

Find the median of $X_3$, and hence state whether the median of $X_3$ is greater than or less than $\mathrm{E}(X_3)$.

By using the substitution $u=1-x$, show that $\mathrm{E}(X_n)={\displaystyle \frac{1}{n+1}}$.

Show that $\mathrm{V}\mathrm{a}\mathrm{r}(X_n)={\displaystyle \frac{n}{(n+1{)}^2(n+2)}}$.

Show that the median of $X_n$ is $m_n=1-2^{-\frac{1}{n}}$.

Show that, for $n\in {\mathbb{Z}}^{+}$, the inequality $\mathrm{E}(X_n)>m_n$ is equivalent to ${(1+{\displaystyle \frac{1}{n}})}^n>2$.

By using the binomial theorem, prove that $\mathrm{E}(X_n)>m_n$ for all integers $n\ge 2$.

Use l'Hôpital's rule to find ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }n{m}_n}$.

Hence find ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }\frac{m_n}{\mathrm{E}(X_n)}}$.

Find the value of $p$ and the value of $q$.

Find $\mathrm{V}\mathrm{a}\mathrm{r}(X)$.

Find $\mathrm{E}(4-3X)$ and $\mathrm{V}\mathrm{a}\mathrm{r}(4-3X)$.

Find the value of $k$.

Use calculus to find the mode of $X$.

Determine whether the median of $X$ is greater than or less than the mode. Justify your answer.

Consider the case $n=6$. Find $\mathrm{E}(X)$ and $\mathrm{V}\mathrm{a}\mathrm{r}(X)$.

Show that, for a spinner with $n$ sectors, $\mathrm{E}(X)={\displaystyle \frac{n+1}{2}}$.

Show that $\mathrm{V}\mathrm{a}\mathrm{r}(X)={\displaystyle \frac{n^2-1}{12}}$.

In a continuous model, the score is replaced by a random variable $Y$ with probability density function

Show that $\mathrm{E}(Y)={\displaystyle \frac{n+1}{2}}$ and $\mathrm{V}\mathrm{a}\mathrm{r}(Y)={\displaystyle \frac{n^2}{12}}$.

The value of $X$ can be obtained by rounding $Y$ to the nearest integer. Show that $\mathrm{V}\mathrm{a}\mathrm{r}(Y)-\mathrm{V}\mathrm{a}\mathrm{r}(X)$ does not depend on $n$, and state its value.

The spinner is replaced by a biased spinner with the same $n$ sectors. Its score $W$ has $\mathrm{P}(W=r)={\displaystyle \frac{2r}{n(n+1)}}$, for $r=1,2,\dots ,n$.

Show that $\mathrm{E}(W)={\displaystyle \frac{2n+1}{3}}$.

Show that $\mathrm{V}\mathrm{a}\mathrm{r}(W)={\displaystyle \frac{(n-1)(n+2)}{18}}$.

A continuous model for $W$ is the random variable $V$ with probability density function

Find $\mathrm{V}\mathrm{a}\mathrm{r}(V)$ in terms of $n$.

Find ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }\frac{\mathrm{V}\mathrm{a}\mathrm{r}(W)}{\mathrm{V}\mathrm{a}\mathrm{r}(V)}}$, and explain briefly the significance of this result.

In a game, a player pays $$c$ to spin the biased spinner with $n=10$ sectors and then receives $$W$.

Find the value of $c$ for which the game is fair.