Radioactive mass A emits particles at a mean rate of 20 per minute, and radioactive mass B emits particles at a mean rate of 25 per minute.
a. What is the probability that fewer than 200 particles are emitted by both masses together in a five-minute time period?
b. What is the probability that mass B emits more particles than mass A in a two-minute time period?
(a) Let X be the number of particles emitted by mass A and let Y be the number of particles emitted by mass B in a five-minute time period. Then X ~ Poisson(100) and Y ~ Poisson(125).
Now X is approximately normal with mean 100 and variance 100, and Y is approximately normal with mean 125 and variance 125.
The number of particles emitted by both masses together is S=X+Y.
Since X and Y are independent, and both approximately normal, S is approximately normal as well, with mean 
and standard deviation 
The z-score of 200 is (200-225)/15=-1.67 The area to the left of z=-1.67 is 0.0475.
(b) Let X be the number of particles emitted by mass A and let Y be the number of particles emitted by mass B in a two-minute time period. Then X~Poisson(40) and Y~ Poisson(50).
Now X is approximately normal with mean 40 and variance 40, and Y is approximately normal with mean 50and variance 50.
The difference between the numbers of particles emitted by the two masses is D= Y?X.
Mass B emits more particles than mass A if D>0.
Since X and Y are independent, and both approximately normal, D is approximately normal as well, with mean
and standard deviation 
The z-score of 0 is (0-10)/9.486833=-1.50. The area to the right of z=-1.05 is 1-.01469=0.8531 .
P(D>0) = 0.8531.
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