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Sales Expense A salesperson's monthly expenses (i…

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Problem 30 Medium Difficulty

Fast-Food Outlets The number of new fast-food outlets opening during June in a certain city is exponentially distributed, with a mean of $5 .$
a. Give the probability density function for this distribution.
b. What is the probability that the number of outlets opening is between 2 and 6$?$

Answer

a. $f(x)=0.2 e^{-0.2 x}$
b. $\approx 0.369$

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Calculus 1 / AB

Finite Mathematics and Calculus with Applications

Chapter 18

Probability and Calculus

Section 3

Special Probability Density Functions

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Video Transcript

Okay, So for this question, we are told that we have, uh, a an exponentially distributed random variable. And we're told to give the probability density function for this variable So half of axe is going to be equal to And we know our probability densely functions for exponential zlook like mamduh e to the negative lamb, that X right where lambda is our rate, we can also be given information in the form of the mean And if we're told what the mean is, that's just one over the rate. Okay? And you can remember that because when we think about our formula in this form, we know that the expectation is one over lambda. Well, one over lambda is just exactly the meat. So if they tell us the mean, we know that we're getting one over of this. So how we would end up writing this since we're told that we have Amina five, we would write this as 1/5 e to the negative x over five. The next part of that asks the probability that X is between two and six probability to less than or equal to, uh, X less than or equal to six. So we are going to integrate from 26 are 1/5. You too. Negative. X over five the X. And so when we do that, we're gonna get negative. E I think it x over five evaluated from two to native six. Playing in our upper bound gives us negative e to the negative 6/5 plus e to the negative 2/5, which I would rewrite just a little bit cleaner as e to the negative 2/5 plus e to the negative 6/5. So that's gonna be our final second part. And our finalists for the first part was just up here the density function.

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