# Poisson distribution - adapting for different intervals

Over a long period of time ,a plumber finds that on average he receives 2 emergency calls per week .Work out the probability of one emergency call on one day assuming the plumber is available for emergency calls five days a week .

• maybe at least one? – quester Oct 20 at 7:39
• At least once gives a probability of about $0.33$. Your answer appears to be right unless you forgot to mention any details in the problem. It's clear that you got your answer by finding $\mathsf P(X=1)$ where $X\sim\mathsf{Pois}\left(\frac{2}{5}\right)$. However, in the future, please show how you got your answer. – Remy Oct 20 at 7:41
• Thanks ,yes will show working in the future ! – Sara Oct 20 at 7:48
• You should still edit to show how you got 0.268 – Glen_b -Reinstate Monica Oct 20 at 11:39

If $$X_5 \sim \mathsf{Pois}(\lambda_5 = 2),$$ and assuming that emergency calls tend to be spread evenly among days, then $$X_1 \sim \mathsf{Pois}(\lambda_1 = 2/5)$$ describes the distribution of emergency calls on any one particular day.

Then using the Poisson PDF function dpois in R, we get $$P(X_1 = 1) =0.2681$$ and (as @remy says) $$P(X_1 \ge 1) = 1 - P(X_1 = 0) = 0.3297.$$

 dpois(1, 2/5)
[1] 0.268128

1 - dpois(0, 2/5)
[1] 0.32968


Here is a bar graph (made with R) of the PDF of $$\mathsf{Pois}(2/5).$$

x=0:5;  pdf=dpois(x, 2/5)
plot(x, pdf, type="h", lwd=3, ylab="PDF", xlab="x",
ylim=c(0,max(pdf)), main="PDF of POIS(2/5)")
abline(v=0, col="green2");  abline(h=0, col="green2")


Note: I've tried finding answers to several plausible (and some not quite so plausible) alternative questions. It may be worthwhile to notice that $$P(X_1 \le 1) = 0.3984.$$ That would be the answer to "at most 1 emergency call on a particular day."

ppois(1, 2/5)
[1] 0.9384481

• So the question should have been atmost 1 . – Sara Oct 21 at 6:15
• Not 'almost', but 'at most'. Not more than 1. $P(X_1 \le 1)$. Question as you have it makes sense. But 'at most 1' gets you the answer claimed. – BruceET Oct 21 at 6:58