A catalog company that receives the majority of its orders by telephone conducte

Question

A catalog company that receives the majority of its orders by telephone conducted a study to determine how long customers were willing to wait on hold before ordering a product. The length of time was found to be a random variable best approximated by an exponential distribution with a mean equal to 3 minutes. Find the waiting time at which only 10% of the customers will continue to hold. - I see the answer is 2.3 minutes but I do not understand the math. could you explain more fully - I am not getting the e math.

Explanation / Answer

Let X = the length of time customers were willing to wait.

Then X ~ Exponential().
Since E(X) = 1/ = 3, we have = 1/3.

Recall that the cdf of X is F(x) = 1 - e^(-x).   ; where cdf is Cumulative Distribution Function, x is time in over case

so we have have to check for 10% customers i.e. 10/100 = 0.1 probability;
Thus:
P(X t)
= 1 - e^(-(1/3)(t)) =1/10, we have to find the value of t

from above

1-0.1 = 0.9 = e^(-(1/3)(t))

i.e. 1/0.9 = e^((1/3)(t))

i.e. ln(1/0.9) = t/3

t = 3 * ln(1/0.9)

which can give you the answer

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