The Poisson distribution with rate of 46 tornados per year h

The Poisson distribution with rate of 4.6 tornados per year has been
proposed as a model for the number of tornados in the state of New
York.

Using appropriate probability model(s), ...

a) What is the probability of no tornados is a given year?

b) What is the probability of no tornados for 3 years in a row?

c) What is the probability of at least 2 tornados in a given year?

d) What is the probability of at most 2 tornados in a given year?

e) In 2013, New York state had 6 tornados - is this unusual? Explain.

f) What are reasonable values for the number of tornados
   in a given year for the state of New York? Explain.

Solution

a) What is the probability of no tornados is a given year?

Given X~Poisson(mean=4.6)

P(X=x)=(4.6^x)*exp(-4.6)/x!

So P(X=0) =(4.6^0)*exp(-4.6)/1=0.01005184

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b) What is the probability of no tornados for 3 years in a row?

0.01005184^3 =0.000001015633

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c) What is the probability of at least 2 tornados in a given year?

P(X>=2)=1-P(X=0)-P(X=1)

=1-(4.6^0)*exp(-4.6)/1 -(4.6^1)*exp(-4.6)/1

=0.9437097

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d) What is the probability of at most 2 tornados in a given year?

P(X<=2)= P(X=0)+P(X=1)+P(X=2)

=(4.6^0)*exp(-4.6)/1 +(4.6^1)*exp(-4.6)/1+(4.6^2)*exp(-4.6)/2

=0.1626387

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e) In 2013, New York state had 6 tornados - is this unusual? Explain.

P(X=6) =(4.6^6)*exp(-4.6)/6! =0.1322696

Since the probability is greater than 0.05, this is not unusual

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f) What are reasonable values for the number of tornados
   in a given year for the state of New York? Explain.

4.6 because it is the mean