Poisson Distribution Exercise

1. Use the Poisson probability formula to find the following probabilities for the distribution X:

a. P(X = 2) when λ = 3

b. P(X = 1) when λ = 0.5

c. P(X = 0) when λ = 1.2

2. Use the Poisson Distribution tables to find the following probabilities:

a. P(X = 5) when λ = 3

b. P(X = 1) when λ = 0.7

c. P(X < 3) when λ = 2

3. A stunt person injures himself an average of three times a year. Use the Poisson probability formula to calculate the probability that he will be injured:

 a. 4 times a year b. Less than twice this year. c. More than three times this year. d. Once in the six months. 4. Occasionally, a machine producing steel tools needs to be reset. The random variable Y is the number of resettings in a month and is modelled by a Poisson distribution. The mean number of resettings needed per month has been found to be 6.
Find the probability that:

a. 7 resettings per month are needed.

b. Fewer than 3 resettings per month are needed.

c. More than 4 resettings per month are needed.

5. The probability that an individual suffers a bad reaction to a injection is 0.002. If 2000 people are injected use the Poisson distribution to find the probability that:

a. Exactly 2 people have a bad reaction.

b. More than 3 people have a bad reaction.

6.

 An ornithologist is studying takahes, a species of bird which nests on the ground. He finds that, on average, there are three nests per hectare in regions where the bird is found. Let the random variable N be the number of nests in a given area. a. N has a Poisson distribution, give a reason to support this. b. If the ornithologist searches an area of one hectare, find the probability that he will find at least one nest. Show your reasoning. c. If he searches an area of two hectares, find the probability that he will find at least one nest. Show your reasoning. d. what area would he need to search in order to have 0.01 probability of finding no nests. 7. A book containing 300 pages has 480 typing errors.

a. What is the average number of errors per page.

b. Find the probability that a page selected at random contains:

(i) No errors
(ii) Exactly 3 errors
(iii) More than two errors

8. The number of calls to the help desk of a company has a Poisson distribution with 36 calls for a 24 hour period. If C = the random variable for the number of calls per hour, find:

a. The probability that the help desk will receive only one call in the first hour.

b. the probabilty that the help desk will receive more than the average number of calls in a particular hour.

c. The mean number of calls per hour E(C).

d. The variance of C.

e. The standard deviation of C.