Problem 12BSC

Use the Poisson distribution to find the indicated probabilities.

Deaths from Horse Kicks A classical example of the Poisson distribution involves the number of deaths caused by horse kicks to men in the Prussian Army between 1875 and 1894. Data for 14 corps were combined for the 20-year period, and the 280 corps-years included a total of 196 deaths. After finding the mean number of deaths per corps-year, find the probability that a randomly selected corps-year has the following numbers of deaths:

(a) 0,

(b) 1,

(c) 2,

(d) 3,

(e) 4.

The actual results consisted of these frequencies: 0 deaths (in 144 corps-years) ;1 death (in 91 corps-years); 2 deaths (in 32 corps-years); 3 deaths (in 11 corps-years); 4 deaths (in 2 corps-years). Compare the actual results to those expected by using the Poisson probabilities. Does the Poisson distribution serve as a good tool for predicting the actual results?

Problem 12BSC

Answer:

Step1 of 3:

We have A classical example of the Poisson distribution involves the number of deaths caused by horse kicks to men in the Prussian Army between 1875 and 1894. Data for 14 corps were combined for the 20-year period, and the 280 corps-years included a total of 196 deaths. After finding the mean number of deaths per corps-year,

We need to find the probability that a randomly selected corps-year has the following numbers of deaths:

(a) 0,

(b) 1,

(c) 2,

(d) 3,

(e) 4.

The actual results consisted of these frequencies: 0 deaths (in 144 corps-years) ;1 death (in 91 corps-years); 2 deaths (in 32 corps-years); 3 deaths (in 11 corps-years); 4 deaths (in 2 corps-years).

Step2 of 3:

Consider the mean number of deaths per corps-year is given by

Consider a random variable “x” follows poisson distribution with sample size “n” and parameter “”.

i,e.X P()

Probability mass function of poisson distribution is given by

P(x) = , x = 0,1,2,...,n.

Where, e = 2.71828.

And

a).

In a randomly selected corps-year,probability that the number of deaths is zero is given by P(x = 0) = ?

Consider,

P(x) =

P(x = 0) =

=

=

= 0.0000554.

b).

In a randomly selected corps-year,probability that the number of deaths is zero is given by P(x = 1) = ?

Consider,

P(x) =

P(x = 1) =

=

=

= 0.00054.

Step3 of 3:

c).

In a randomly selected corps-year,probability that the number of deaths is zero is given by P(x = 2) = ?

Consider,

P(x) =

P(x = 2) =

=

=

= 0.002.

d).

In a randomly selected corps-year,probability that the number of deaths is zero is given by P(x = 3) = ?

Consider,

P(x) =

P(x = 3) =

=

=

= 0.0089.

e).

In a randomly selected corps-year,probability that the number of deaths is zero is given by P(x = 4) = ?

Consider,

P(x) =

P(x = 4) =

=

= 0.02131.

Therefore, “yes”, Poisson distribution serve as a good tool for predicting the actual results.