Exponential distribution The occurrence of random events (such as phone calls
or e-mail messages) is often idealized using an exponential distribution. If
\(\lambda\) is the average rate of occurrence of such an event, assumed to be
constant over time, then the average time between occurrences is
\(\lambda^{-1}\) (for example, if phone calls arrive at a rate of \(\lambda=2 /
\mathrm{min}\), then the mean time between phone calls is \(\lambda^{-1}=1 / 2\)
min). The exponential distribution is given by \(f(t)=\lambda e^{-\lambda t},\)
for \(0 \leq t<\infty\).
a. Suppose you work at a customer service desk and phone calls arrive at an
average rate of \(\lambda_{1}=0.8 / \mathrm{min}\) (meaning the average time
between phone calls is \(1 / 0.8=1.25 \mathrm{min}\) ). The probability that a
phone call arrives during the interval \([0, T]\) is \(p(T)=\int_{0}^{T}
\lambda_{1} e^{-\lambda_{1} t} d t .\) Find the probability that a phone call
arrives during the first \(45 \mathrm{s}(0.75 \mathrm{min})\) that you work at
the desk.
b. Now suppose walk-in customers also arrive at your desk at an average rate
of \(\lambda_{2}=0.1 /\) min. The probability that a phone call and a customer
arrive during the interval \([0, T]\) is $$p(T)=\int_{0}^{T} \int_{0}^{T}
\lambda_{1} e^{-\lambda_{1} t} \lambda_{2} e^{-\lambda_{2} s} d t d s$$ Find
the probability that a phone call and a customer arrive during the first 45 s
that you work at the desk.
c. E-mail messages also arrive at your desk at an average rate of
\(\lambda_{3}=0.05 /\) min. The probability that a phone call \(a n d\) a customer
and an e-mail message arrive during the interval \([0, T]\) is
$$p(T)=\int_{0}^{T} \int_{0}^{T} \int_{0}^{T} \lambda_{1} e^{-\lambda_{1} t}
\lambda_{2} e^{-\lambda_{2} s} \lambda_{3} e^{-\lambda_{1} u} d t d s d u$$
Find the probability that a phone call and a customer and an e-mail message
arrive during the first 45 s that you work at the desk.
Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine 
