Question

suppose the number of deaths of a certain cockroach population occur independently at a rate of 3 deaths/day

i) find the probability of having 3 cockroach deaths in a day

ii) if we consider a 20 day sample period, what is the probability that at most 2 of these days have less than 3 deaths?

iii) if we consider a random sample of 42 days, what is the probability that the average number of deaths/day is less than 2.5?

Answer 1

The number of deaths distribution per day here is given as:

X ~ Poisson(3)

a) The probability of having 3 cockroach deaths in a day is computed using poisson distribution probability function as:

Therefore 0.2240 is the required probability here.

b) The probability of less than 3 deaths in a day is computed as:

-3 P(X <3) = e-3 + 3e-3 + 3 e = 0.4232

The probability that at most 2 of these days will have less than 3 deaths is computed here using binomial probabiltiy function as:

(20) 18 =(1-0.4232) + 20 +0.4232 *(1-0.4232) + 10.423271-0.4232) 2

0.0020

Therefore 0.0020 is the required probability here.

c) For 42 days, the probability that the average number of deaths / day is less than 2.5 is computed here as:

P(X < 2.5

Converting it to a standard normal variable, we have here:

2.5 - 3 P(Z < 3 42

P(Z < -1.87)

Getting it from the standard normal tables, we have here:

P(Z < -1.87) = 0.0307

Therefore 0.0307 is the required probabiltiy here.