Question

A major flooding in a given year has a Poisson distribution with a mean occurrence of 2.5

d) How many months have to pass to be in the 80th percentile?

Thank you!!

Answer 1

We have to use the Poisson distribution table to find the answer.

Let x be the number of months has major flooding incidents .

We are given $\lambda$ = 2.5 and asked to find x such that sum of probabilities up to that x is 0.8

So first we need probabilities for each x with $\lambda$ = 2.5

2.6 2.1 2.2 2.3 2.4 2.5 2.7 2,8 2.9 3.0 .1003 0821 .1225 .1108 .0907 0743 0672 .0608 .0550 .0498 2572 2438 2306 2177 2052 .1931 .1815 1703 .1596 .1494 2565 2 2700 2681 2652 2613 2510 2450 2384 2314 2240 .1890 .1966 2033 2090 2138 2176 2205 2225 2237 .2240 1336 0992 .1082 1169 .1254 1414 .1488 1557 1622 1680 0735 .0940 .1008 0417 .0476 0538 0602 0668 .0804 .0872 6 .0146 0174 .0206 0241 .0278 .0319 .0362 .0407 .0455 .0504 0068 0083 7 0044 0055 0099 .0118 0139 .0163 0188 0216 0011 0015 .0019 0025 0031 0038 .0047 .0057 0068 .0081 0009 .001 0014 .0018 0022 .0027 0003 .0004 0005 0007 10 .0001 0001 .0001 0002 0002 .0003 0004 .0005 0006 .0008 11 0000 .0000 0000 0000 .0000 0001 0001 .0001 0002 .0002 12 .0000 .0000 .0000 0000 0000 .0000 .0000 .0000 .0000 .0001

So first we need to find cumulative probabilities for each x ..So we goes on adding the each probability to its previous probability

The cumulative for x = 0 would be P(X) = 0.0821

The cumulative for x = 1 would be 0.0821 + 0.2052 = 0.2873

The cumulative for x = 2 would be 0.2873 + 0.2565 = 0.5438

Similarly we have to find cumulative for all x

P(X) Cumulative X 0 0.0821 0.0821 1 0.2052 0.2873 2. 0.2565 0.5438 3 0.2138 0.7576 4 0.1336 0.8912 0.0668 0.958 6 0.0278 0.9858 7 0.0099 0.9957 8 0.0031 0.9988 0.0009 0.9997 10 0.0002 0.9999 11 0 0.9999 12 0 0.9999 LO

Since the cumulative for x = 4 is 0.8912 , so it exceeds 0.8 at x = 4 first time

The required number of months would be 3

Because the percentile or sum of probabilities up to x = 3 is approximately equal to 0.80

Therefore approximately 3 months have to pass to be in the 80th percentile