16. the design of a dam, the occurrence of floods is modeled with the Poisson di

Question

16. the design of a dam, the occurrence of floods is modeled with the Poisson distri tion function. The occurrence rate for the floods is mentioned as 0.01. However, further investigations show that the occurrence rate may be as high as 0.02. Since there are merits with both estimates, we decide that the two occurrence rates are both likely with 50-50 chances. (a) Compute the probability that the dam will not experience any floods in the next 10 (b) Compute the probability that the dam will experience no more than two floods in the next 10 yr

Explanation / Answer

Solution

Back-up Theory

If a random variable X ~ Poisson(), i.e., X has Poisson Distribution with mean then

probability mass function (pmf) of X is given by P(X = x) = e – .x/(x!) …………..(1)

where x = 0, 1, 2, ……. ,

Values of p(x) for various values of and x can be obtained by using Excel Function.

If X = number of times an event occurs during period t, Y = number of times the same event occurs during period kt, and X ~ Poisson(), then Y ~ Poisson (k) …………….. (2)

Now, to work out solution,

Let X = number of floods experienced by the dam in a year. Then, we are given that

X ~ Poisson(0.01) with probability ½ and X ~ Poisson(0.02) with probability ½.

Further, if Y = number of floods experienced by the dam in 10 years, then

[vide (2) under Back-up Theory], Y~ Poisson(0.1) with probability ½ and Y ~ Poisson(0.2) with probability ½.

All probabilities are found by Excel Function

Q1 Part (a)

Probability dam will not experience any flood in next 10 years = P(Y = 0)

= (½){P(Y = 0/ = 0.1) + P(Y = 0/ = 0.2)} = (½)(0.9048 + 0.8187) = 0.8618 ANSWER

Q1 Part (b)

Probability dam will experience no more than 2 flood in next 10 years = P(Y 2)

= (½){P(Y 2 / = 0.1) + P(Y 2/ = 0.2)} =(½)(0.9998 + 0.9989) = 0.9994 ANSWER

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