Question

Suppose that buses are scheduled to arrive at a bus stop at noon but are always X
minutes late, where X is an exponential random variable with probability density function
fX(x) = ex: Suppose that you arrive at the bus stop precisely at noon.

Answer 1

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Looking at other sources :- Complete Question:

Suppose that the buses arrive are scheduled to arrive at a bus stop at noon are always X minutes late, where X is an exponential random variable with probability density function $\sf \red{ f_X(x)=\lambda e^{-\lambda x} , x \geq 0}$ ,Suppose you arrive at the bus stop precisely at noon . Find the probability that you have to wait for more than 5 min for the bus to arrive.

1) We need to find

$\begin{gathered} \sf \: P(x\geq 5)=\int\limits^{\infty}_5 {f_X(x)} \, dx \\ \\ \sf=\int\limits^{\infty}_5 {\lambda e^{-\lambda x} } \, dx \\ \\ \sf =\bigg\rvert_5^\infty \frac{\lambda e^{-\lambda x}}{-\lambda} \\ \\ \sf= \bigg\rvert_5^\infty -e^{-\lambda x}\\ \\ \sf \red{= e^{-5\lambda}}\end{gathered}$

Above answer is our required probability.