The Exponential Distribution and Exponential Random Variables | Probability Theory

What is the exponential distribution? This is one of the most common continuous probability distributions. We'll go over an introduction of the exponential distribution and exponentially distributed random variables in today's probability theory video lesson.

The exponential distribution is often used to model the time until a particular event occurs, or the time between events. We'll see a couple examples of this in the lesson.

Remember that continuous probability distributions have probability density functions (PDF), which don't give actual probabilities. The PDF of the exponential distribution is f(x) = \lambda*e^{-\lambda*x} when x is greater than or equal to 0. Note that "\lambda" is our way of writing the greek character lambda here in a YouTube description where we cannot actually write lambda. When x is less than 0, f(x) = 0, so there is no probability in an exponential distribution for negative values of x.

More interesting than the PDF of a continuous probability distribution is the cumulative distribution function (CDF), which gives actual probabilities! Let X be an exponential distributed random variable with parameter lambda. The CDF of the exponential distribution, evaluated at a, is 1 - e^{-\lambda*a}. This is the probability that X is less than or equal to a. We'll prove the CDF of the exponential distribution using the definition of CDF in this lesson, and we'll go over some CDF example problems!

We also go over the expected value, or mean, of an exponential distribution, as well as the variance and the standard deviation. 

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SOLUTION TO PRACTICE PROBLEM:

Let X be the amount of time a husband spends shopping for an anniversary card. Thus, X is exponential with mean 9 minutes. Hence, it's parameter is the inverse: 1/9. So the pdf is, for x greater than or equal to 0, f(x) = -(1/9)*e^{-(1/9)*x}. 

The CDF is F(a) = 1 - e^{-(1/9)*a}.

The probability that a husband spends between 2 and 4 minutes shopping for an anniversary card is the probability that X is between 2 and 4, which is F(4) - F(2) = ( 1 - e^{-(4/9)} ) - ( 1 - e^{-(2/9)} ) = e^{-2/9} - e^{-4/9} which is roughly 16%. 

This problem was adapted from Exercise 5.3.1 from this source: https://stats.libretexts.org/Bookshel...
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The example problem we went over in the video was adapted from "A First Course in Probability Theory" by Sheldon Ross, the first textbook I used to learn probability theory! Check out the book and see if it suits your needs! You can purchase the textbook using the affiliate link below which costs you nothing extra and helps support Wrath of Math!

PURCHASE THE BOOK: https://amzn.to/31mXEjr

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Vallow Bandcamp: https://vallow.bandcamp.com/
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