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My textbook has this exercise, in the section regarding exponential distribution:

Given an arrival process with $\lambda = 8.0$, what is the probability that an arrival occurs in the first $t = 7$ time units?

I used this formula:

$F(t)=1-e^{-λt}$

To give this solution:

$P=1-e^{(-8)(7)}=1-e^{-56}\approx1$

But the suggested solution was this:

$P=1-e^{(-\frac18)(7)}=1-e^{-\frac78}\approx0.58314$

Could you kindly explain me why I was wrong? I really don't get why the book used the mean instead of using lambda in the formula. To me it made sense that, if there's an average of 8 arrivals every $t$, the probability of a single arrival occurring in the first 7 $t$ should be almost 1.

Thanks!

alkazam
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    "$\lambda$" without context is meaningless. You need to find the definition of "arrival process" used by your reference and apply it as given. Most likely its "$\lambda$" is meant be be used in the form $t/\lambda$ rather than $\lambda t$ in the formulas. – whuber Mar 08 '19 at 12:06
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    R statistical software uses rate $\lambda$ as parameter; its CDF function pexp(7, 1/8) returns 0.583138. Here, the parameter (second argument) is $\lambda = 1/8.$ . – BruceET Mar 09 '19 at 10:32
  • If $X$ has the distribution $X \sim \mathsf {EXP}(μ=8)$$≡ \mathsf{EXP}(λ=1/8),$ then $P(X≤7)$$=1−e^{−7/8}$$=0.583138.$ If the time unit is hours then $μ=8$ means 8 events in an hour on average and $λ=1/8$ means an event occurs about every eighth of an hour or 7.5 minutes. I can see how someone--speaking loosely--might say that the "rate" is about 8 in an hour (to mean an average of 8 events in an hour), but the usual technical use of the word rate has units of "per hour" (reciprocal hours) and it means an event every eighth of an hour. – BruceET Mar 09 '19 at 11:09
  • Please add the [tag:self-study] tag & read its wiki. Then tell us what you understand thus far, what you've tried & where you're stuck. We'll provide hints to help you get unstuck. Please make these changes as just posting your homework & hoping someone will do it for you is grounds for closing. – kjetil b halvorsen Nov 04 '21 at 23:27

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