The time interval between the arrivals of successive trains at a certain station is measured by a random variable X with a probability density function $f ( x ) = \left\{ \begin{array} { l l }
0.2 e ^ { - 0.2 x } & \text { for } x \geq 0 \
0 & \text { for } x

Question

The time interval between the arrivals of successive trains at a certain station is measured by a random variable X with a probability density function $f ( x ) = \left\{ \begin{array} { l l } 
0.2 e ^ { - 0.2 x } & \text { for } x \geq 0 \
0 & \text { for } x < 0
\end{array} \right.$ where x is the time (in minutes)between the arrivals of a randomly selected pair of successive trains.What is the probability that two successive trains selected at random will arrive within 9 minutes of one another? Round to the nearest hundredth.
A)0.17
B)0.83
C)0.62
D)0.42

Quizplus · Accepted Answer

The given probability density function $f(x) = 0.2e^{-0.2x}$ for $x \geq 0$ represents an exponential distribution with a rate parameter $\lambda = 0.2$. To find the probability that the time between arrivals is less than 9 minutes, we integrate the density function from 0 to 9:$$P(X < 9) = \int_{0}^{9} 0.2e^{-0.2x} dx$$This evaluates to:$$= -e^{-0.2x} \Big|_{0}^{9}$$$$= -(e^{-0.2 \cdot 9} - e^{0})$$$$= 1 - e^{-1.8}$$Using a calculator or a table for the exponential function:$$= 1 - e^{-1.8} \approx 1 - 0.1653 \approx 0.8347$$Rounded to the nearest hundredth, this gives a probability of 0.83.

The Time Interval Between the Arrivals of Successive Trains at a Certain