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I have a question on the exponential distribution:

Question

The waiting time to receive food after placing an order at the local sandwich shop follows an exponential distribution with a mean of 70 seconds. 80 percent of the patrons wait less than how many minutes to receive their food?

Attempt

My attempt was to assume $\lambda$ as $7/6$ minute per order. However, I am clueless about how to move on. I attempted to use the equation $F(x) = 1 - e^{-\lambda x}$ with result to be $0.61$, which differs greatly from the model answer $1.878$ minutes.

Appreciate some guidiance as to what I am not doing right please. Is the equation used even correct and if $x$ should be $0.8$?

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3 Answers 3

up vote 5 down vote accepted

The cumulative distribution function expresses $P(X \le x)$ as a function of $x$. Your interest is in finding the value of $x$ such that the CDF is equal to 0.8. So, begin by setting the $P(X \le x) = 0.8$. Just make sure you correctly set $\lambda$, noting that the mean of the exponential distribution is $\lambda^{-1}$:

$0.8 = 1-e^{-x/70}$

Solving for $x$ gives you the right answer in seconds, then convert to minutes.

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Set $\lambda=1/70$. The exponential survival function is $\exp(-\lambda x)$, so set $\exp(-x/70)=1-0.8=0.2$, by goal seeker you get $112,5$ seconds.

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After taking the advice from above, I have formed up the following equations. Appreciate some advice if I am doing it right.

enter image description here

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1  
It is correct. Compare this to the median of the service time (50% of the customers), which is 70sec*ln(2) = 48.5 secs. –  Hernan Apr 28 at 6:56

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