TU Wien:Statistik und Wahrscheinlichkeitstheorie UE (Levajkovic)/Übungen 2023W/HW05.2

Mail order company

A mail order company provides free examination of its products for 7 days. If not completely satisfied, a customer can return the product within that period and get a full refund. According to past records of the company, an average of 2 of every 10 products sold by this company are returned for a refund.

(a) Compute the probability that no more than 6 of the 40 products sold by this company on a given day will be returned for a refund.

(b) Use a Poisson distribution to approximate the probability in (a). What can be said about the accuracy of this approximation?

Dieses Beispiel ist als verified_by_tutor markiert. Ist dies falsch oder ungenau? Aktualisiere den Lösungsstatus (Details: Vorlage:Beispiel)

Lösung von Tutorin

(a)

Let $X$ be the number of products sold by this company on a given day that are returned for a refund. Then, $X\sim bin(40,2/10)$. The goal is to compute $P(X\leq 6)$. In other words, the probability of success (that is, the probability that a product is returned) is $p=0.2$ and the number of trials (products sold) is $n=40$. We are to find the probability of at most six successes (returns).

$$P(X\leq 6)={\texttt {pbinom(6,40,0.2)}}=0.2858914.$$

The expectation $\mathbb {E} X=np=8$ and variance $\mathbb {V} arX=np(1-p)=6.4$.

(b)

Since $\mathbb {E} X=8$ we approcimate $X$ by a random variable $Y\sim {\mathcal {P}}(8)$. Then,

$$P(X\leq 6)\approx P(Y\leq 6)={\texttt {ppois(6,8)}}=0.3133743.$$

The (exact) probability is computed directly by using the binomial distribution $bin(40,0.2)$ and is 0.2858914, while the approximate value, when applying the Poisson distribution with $\lambda =8$, is 0.3133743. The error of the approximation is about 2.7%.