Mathematics 9709 · AS & A Level · Continuous random variables

Continuous random variables — practice question

The durations needed to perform Beethoven’s Sixth Symphony may be regarded as following a normal distribution with mean $41.1$ minutes and standard deviation $3.4$ minutes. Three performances of this symphony are selected at random.
(i)[4]

Determine the probability that, out of these occasions, the symphony lasts more than $42$ minutes exactly $1$ time.

(ii)[5]

The times needed to play Beethoven’s Fifth Symphony may also be treated as normally distributed. The chance that the time is below $26.5$ minutes is $0.1$, while the chance that the time exceeds $34.6$ minutes is $0.05$. Find the mean and standard deviation of the times taken to play this symphony.

(iii)[4]

Given that the times to perform the two symphonies are independent, find the probability that, when both symphonies are performed, both times are under $34.6$ minutes.

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