Question

Data from 14 cities were combined for a 20-year period, and the total 280 city-years included a total of 107 homicides. After finding the mean number of homicides per city-year, find the probability that a randomly selected city year has the following numbers of homicides, then compare the actual results to those expected by using the Poisson probabilities a. P(0) = (Round to four decimal places as needed.) b. P(1) = (Round to four decimal places as needed.) c, P(2) = (Round to four decimal places as needed.)d. P(3) = (Round to four decimal places as needed.)

Answer 1

Answer:

a) P(0) = 0.6824

b) P(1) = 0.2608

c) P(2) = 0.0498

d) P(3) = 0.00635

Step-by-step explanation:

Number of units (n) in the data, n = 280

Total number of events, k = 107

The probability using Poisson probabilities is calculated using the formula;

P(x) = $\frac{e^{-\lambda}\lambda^x}{x!}$

here,

Poisson parameter, λ = $\frac{107}{280}$

Thus,

a) P(0) = $\frac{e^{-\frac{107}{280}}(\frac{107}{208})^0}{0!}$

or

P(0) = 0.6824

b) P(1) = $\frac{e^{-\frac{107}{280}}(\frac{107}{280})^1}{1!}$

or

P(1) = 0.2608

c) P(2) = $\frac{e^{-\frac{107}{280}}(\frac{107}{280})^2}{2!}$

or

P(2) = 0.0498

d) P(3) = $\frac{e^{-\frac{107}{280}}(\frac{107}{280})^3}{3!}$

or

P(3) = 0.00635