Question

This cumulative review problem uses material from Chapters 3, 5, and 10. Recall that the Poisson distribution deals with rare events. Death from the kick of a horse is a rare event, even in the Prussian army. The following data are a classic example of a Poisson application to rare events. The data represent the number of deaths from the kick of a horse per army corps per year for 10 Prussian army corps over a period of time. Let x represent the number of deaths and f the frequency of x deaths.

x	0	1	2	3 or more
f	105	62	29	4

(a) First, we fit the data to a Poisson distribution.

The Poission distribution states P(x) =

e−λλx

x!

, where

λ ≈ x

(sample mean of x values). From our study of weighted averages, we get the following.x =

Σxf

Σf

Verify that

x = 0.66.

Hint: For the category 3 or more, use 3.

x =

(b) Now we have P(x) =

e−0.66(0.66)x

x!

for

x = 0, 1, 2, 3 .

Find

P(0),

P(1),

P(2),

and

P(3 ≤ x).

Round to three places after the decimal.

P(0)	=
P(1)	=
P(2)	=
P(3 ≤ x)	=

(c) The total number of observations is

Σf = 200.

For a given x, the expected frequency of x deaths is

200P(x).

The following table gives the observed frequencies O and the expected frequencies

E = 200P(x).

x	0 = f	E = 200P(x)
0	105	200(0.517) = 103.4
1	62	200(0.341) = 68.2
2	29	200(0.113) = 22.6
3 or more	4	200(0.029) = 5.8

Compute χ² = Σ 

(O − E)2

E

using the values in the table. (Round your answer to two decimal places.)

Answer 1

Given, 105 62 29 3 oe motė X no Poisson (Mean () Exif Coxso5)+(2 12) +(2 xc2) + (4x3) Mean (x): %3D |05+62+29+4 13 2 %3D 200 Mean(x) = 0.66 %3D Xo Poisson (^=0.66) it

0.66 P(X =0) = O -517 0.66 (0.6)". P(X=1) 0.341 1! -0. 66 c0-66)² =| 0.113 P(X= ) 2! -0- 66 Co.66) P(X =3) 0.025 8! 1- P(x<<) - 1-{r0)+ PO+P@] = 1-0517+0.341 + 0.113] - 1-[0.971] P(34 x) P(x »,3) 0.029

P () = 0.517 P() = 0.34) P(2) = 0.113 P(3<x) = 0.029 E = 200 P(1) %3D %3D 103.4 105 68.2 62 22.6 29 5.8 x?- } (6;- i)*. (1os- 103-4* (c2-68-2)*, (29-22.9 (105- 103.4) (62-68.2)* (29-22.6)* 103.44 22.6 68.2 !3 (4-5.8)2 5.8 = 2.959 4 2.96