$\require{\cancel}$ $\require{\stix[upint]}$

Qtn No.	1	2	3	4	5	6	Total
Marks	8	8	8	8	8	9	49
Score

Question 1 Code: 9709/71/M/J/10/5, Topic: -

The random variable $T$ denotes the time in seconds for which a firework burns before exploding. The probability density function of $T$ is given by

$$ \mathrm{f}(t)= \begin{cases}k \mathrm{e}^{0.2 t} & 0 \leqslant t \leqslant 5 \\ 0 & \text { otherwise }\end{cases} $$

where $k$ is a constant.

$\text{(i)}$ Show that $\displaystyle k=\frac{1}{5(\mathrm{e}-1)}$. $[3]$

$\text{(ii)}$ Sketch the probability density function. $[2]$

$\text{(iii)}$ $80 \%$ of fireworks burn for longer than a certain time before they explode. Find this time. $[3]$

Question 2 Code: 9709/72/M/J/10/5, Topic: -

Question 3 Code: 9709/73/M/J/10/5, Topic: -

The time, in minutes, taken by volunteers to complete a task is modelled by the random variable $X$ with probability density function given by

$$ f(x)= \begin{cases}\dfrac{k}{x^{4}} & x \geqslant 1 \\ 0 & \text { otherwise }\end{cases} $$

$\text{(i)}$ Show that $k=3$. $[2]$

$\text{(ii)}$ Find $\mathrm{E}(X)$ and $\operatorname{Var}(X)$. $[6]$

Question 4 Code: 9709/71/O/N/10/5, Topic: -

The marks of candidates in Mathematics and English in 2009 were represented by the independent random variables $X$ and $Y$ with distributions $\mathrm{N}\left(28,5.6^{2}\right)$ and $\mathrm{N}\left(52,12.4^{2}\right)$ respectively. Each candidate's marks were combined to give a final mark $F$, where $F=X+\frac{1}{2} Y$.

$\text{(i)}$ Find $\mathrm{E}(F)$ and $\operatorname{Var}(F)$. $[3]$

$\text{(ii)}$ The final marks of a random sample of 10 candidates from Grinford in 2009 had a mean of 49. Test at the $5 \%$ significance level whether this result suggests that the mean final mark of all candidates from Grinford in 2009 was lower than elsewhere. $[5]$

Question 5 Code: 9709/72/O/N/10/5, Topic: -

Question 6 Code: 9709/73/O/N/10/5, Topic: -

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

$$ \mathrm{f}(x)= \begin{cases}\frac{1}{6} x & 2 \leqslant x \leqslant 4 \\ 0 & \text { otherwise }\end{cases} $$

$\text{(i)}$ Find $\mathrm{E}(X)$. $[3]$

$\text{(ii)}$ Find the median of $X$. $[3]$

$\text{(iii)}$ Two independent values of $X$ are chosen at random. Find the probability that both these values are greater than 3. $[3]$