Question

Assume that the length of a phone call (minutes) is a continuous random variable X with probability density function {f(x)=1/10e^-0.1x} . If someone arrives at a phone booth just before you arrive, find the probability that you have to visit

1). Less the 5 minutes

2). Greater than 5 minutes

3). Between 5 and 20 minutes

Answer 1

We are given that $f(x) = 0.1e ^{−0.1x} , x ≥ 0$ .

1)

$$P(X<5)=\displaystyle\int_{0}^50.1e ^{−0.1x}dx=[-e ^{−0.1x}]\begin{matrix} 5 \\ 0 \end{matrix}$$

$$=1-e^{-0.5}\approx0.3935$$

2)

$$P(X>5)=\displaystyle\int_{5}^{\infin}0.1e ^{−0.1x}dx=[-e ^{−0.1x}]\begin{matrix} \infin \\ 5 \end{matrix}$$

$$=e^{-0.5}\approx0.6065$$

3)

$$P(5<X<20)=\displaystyle\int_{5}^{20}0.1e ^{−0.1x}dx=[-e ^{−0.1x}]\begin{matrix} 20 \\ 5 \end{matrix}$$

$$=e^{-0.5}-e^{-2}\approx0.4712$$