Question

An article suggests that a Poisson process can be used to represent the occurrence of structural loads over time. Suppose the mean time between occurrences of loads is 0.5 year.

(a) How many loads can be expected to occur during a 4-year period?
loads

(b) What is the probability that more than nine loads occur during a 4-year period? (Round your answer to three decimal places.)


(c) How long must a time period be so that the probability of no loads occurring during that period is at most 0.1? (Round your answer to four decimal places.)
yr

You may need to use the appropriate table in the Appendix of Tables to answer this question.

Answer 1

Concepts and reason

The Poisson distribution is used to calculate the probability of an event occurring over a certain interval.

Example: Count the number of phone calls arriving at a switchboard between 9 and 10 A.M.

Fundamentals

The probability mass function of the random variable can be defined as,

Here, Euler’s constant 2.718

Expected value of the variable

= number of success for the event.

If events in a Poisson process occur at a mean rate of per unit, then the expected number of occurrences in an interval of length is. The probability mass function of the Poisson distribution can be defined as,

The mean of the Poisson distribution is $E(X)=1 $

(a)

Let X be the number of loads occur.

The number of loads can be expected to occur during a 4-year period is,

(b)

The probability that more than nine loads occur during a 4-year period is,

(c)

Here we have to find that, how long a time period must be, so that the probability of no loads occurring during that period is at most 0.1

The occurrences of loads

We have to find the value of time period t at which

Ans: Part a

The number of loads can be expected to occur during a 4-year period is 8.

Part b

The probability that more than nine loads occur during a 4-year period is 0.2834.

Part c

The time period is 1.1513 years.