Answer to Question #189578 in Statistics and Probability for Mariam

Question

Suppose that on average 2 people in a major city die each year from alien attack. Suppose that each

attack is random and independent.

(a) If X is the number of deaths from alien attack within the next year from a randomly selected

major city, what type of random variable is X?

(b) Use the Poisson approximation to approximate the probability that the next major city you visit

will have at least 3 deaths due to alien attack?

(c) Why do you think a Poisson approximation is used instead of a normal approximation?

Answer 1

a.) Suppose that n is the size of the population of the major city and $p = \dfrac{2}{n}$ is the probability

that a randomly selected person drawn from that city is killed by alien attack. The total possible

outcomes of X, i.e. the state space of X, is $S_X = {0, 1, 2, ..., n}$ and,

$P(X=r) = ^nC_r(p)^r(1-p)^{n-r}$

b.)$P(X \ge 3) = 1-P(X<3)$

$= 1-(P(X=0)+P(X=1)+P(X=2))$

$= 1-\dfrac{e^{-2}}{1}- \dfrac{2e^{-2}}{1} -\dfrac{4e^{-2}}{2}$ $= 1-5e^{-2}$

c.) In this scenario, n is quite large compared to np so the Poisson approximation seems like a

good fit. Moreover, np = 2 stays fixed and small while n is quite large and hence np(1 − p) < np <<n

appears quite small with respect to n, which makes the normal approximation not as appealing.