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### ALEEZA

#### Cambridge International AS and A Level

 Name of student ALEEZA Date Adm. number Year/grade 1986 Stream Aleeza Subject Probability & Statistics 2 (S2) Variant(s) P71, P72, P73 Start time Duration Stop time

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

Get Mathematics 9709 Topical Questions (2010-2021) for $14.5 per Subject. Attempt all the 6 questions Question 1 Code: 9709/71/M/J/10/5, Topic: - The random variable$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]\$

Worked solutions: P1, P3 & P6 (S1)

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