$S$ and $T$ are independent random variables with the distributions $\mathrm{N}(18, 3^2)$ and

Question

$S$ and $T$ are independent random variables with the distributions $\mathrm{N}(18, 3^2)$ and $\mathrm{N}(\mu, \sigma^2)$ respectively. It is given that $\mathrm{P}(T < 4) = \mathrm{P}(T > 9)$ and $\mathrm{P}(S < 3T) = 0.65$. Calculate the values of $\mu$ and $\sigma$.[4] A fruit stall sells grapes that are packed in packets with masses in grams that follow the distribution $\mathrm{N}(850, 30^2)$. The grapes are sold at $\$18$ per kilogram. Find the probability that a customer pays more than $\$30$ for two packets of grapes.[2] The fruit stall accepts payment by cash or PayNow. The number of customers who pay by PayNow in a day is a random variable with mean $12$ and variance $4.8$. In a month of $30$ days, find the probability that the average number of customers per day who pay by PayNow is more than $12.3$.[3]

Accepted Answer

(a) $\mu=6.5$, $\sigma=0.827$ (3 s.f.) (b)(i) $0.784$ (3 s.f.) (ii) $0.227$ (3 s.f.)

2024 VJC P2 Q9

2024 VJC P2 Q9

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