Question 5 [13 marks]

A nuclear plant releases a detectable amount of radioactive gases twice a month on average.

Let be the Poisson random variable representing the number of radioactive emissions per month.

Answer all the following questions. Give your final numerical answers correct to 2 decimal places, unless otherwise stated.

1. What is the probability that
2. there are exactlyone radioactive emission in one month? [2 marks]
3. there are at most 1 radioactive emission in one week? Assume that there are 4 weeks in one month. [2 marks]

1. there are at least 12 radioactive emissions in a 3-month period? [2 marks]

1. Let represent the length of time (in weeks) between successive radioactive emissions of the nuclear plant in part (a). is therefore exponentially distributed.
   1. Calculate the parameter value of the distribution of . [2 marks]
   2. Calculate the probability that is between 3 and 7 weeks. [2 marks]

1. Given that it took more than 3 weeks for the first radioactive emission to occur, what is the probability that the next emission would occur within the next week? [3 marks]