Question

Let the continuous r.v. X denote the weight (in pounds) of a package. The range of weight of packages is

between 45 and 60 pounds.

(a) Determine the probability that a package weighs more than 50 pounds.

(b) Find the mean and the variance of the weight of packages.

Answer 1

(a). Probability that the pack weighs more than

 $\begin{aligned} P(x>50) &=1-P(x \leq 50) \\ &=1-P(45<x \leq 50) \\ &=1-\int_{45}^{50} \frac{1}{15} \cdot d x . \\ &=1-\frac{1}{15}(50-45) \\ &=1-\frac{5}{15}=1-\frac{1}{3}=\frac{2}{3} . \end{aligned}$

$\begin{aligned} \text { (b). Mean }&= \mu=\frac{(b+a)}{2}=E(x) . \\ \therefore &\ E(x)=\left(\frac{60+45}{2}\right)=52.5 \\ \text { Variance }=& E\left(x^{2}\right)-[E(x)]^{2} \\ &E\left(x^{2}\right)=\int_{a}^{b} x^{2} \cdot f(x) \cdot d x=\frac{b^{2}+a b+a^{2}}{3} \\ &= \frac{60^{2}+60 \times 45+45^{2}}{3 .} \\ \therefore \text { Variance }=& 2775-(52.5)^{2} \\ \text { Variance }=18.75 \end{aligned}$