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Qtn No.	1	2	3	Total
Marks	8	11	9	28
Score

Question 1 Code: 9709/61/M/J/11/5, Topic: The normal distribution

$\text{(a)}$ The random variable $X$ is normally distributed with mean $\mu$ and standard deviation $\sigma$. It is given that $3 \mu=7 \sigma^{2}$ and that $\mathrm{P}(X>2 \mu)=0.1016$. Find $\mu$ and $\sigma$. $[4]$

$\text{(b)}$ It is given that $Y \sim \mathrm{N}(33,21)$. Find the value of $a$ given that $\mathrm{P}(33-a$[4]$

Question 2 Code: 9709/63/M/J/11/5, Topic: The normal distribution

The random variable $X$ is normally distributed with mean $\mu$ and standard deviation $\frac{1}{4} \mu$. It is given that $\mathrm{P}(X>20)=0.04$.

$\text{(i)}$ Find $\mu$. $[3]$

$\text{(ii)}$ Find $\mathrm{P}(10 < X < 20)$. $[3]$

$\text{(iii)}$ 250 independent observations of $X$ are taken. Find the probability that at least 235 of them are less than 20. $[5]$

Question 3 Code: 9709/62/O/N/11/5, Topic: Discrete random variables

A triangular spinner has one red side, one blue side and one green side. The red side is weighted so that the spinner is four times more likely to land on the red side than on the blue side. The green side is weighted so that the spinner is three times more likely to land on the green side than on the blue side.

$\text{(i)}$ Show that the probability that the spinner lands on the blue side is $\frac{1}{8}$. $[1]$

$\text{(ii)}$ The spinner is spun 3 times. Find the probability that it lands on a different coloured side each time. $[3]$

$\text{(iii)}$ The spinner is spun 136 times. Use a suitable approximation to find the probability that it lands on the blue side fewer than 20 times. $[5]$