Expected Value of the Exponential Distribution | Exponential Random Variables, Probability Theory

What is the expected value of the exponential distribution and how do we find it? In today's video we will prove the expected value of the exponential distribution using the probability density function and the definition of the expected value for a continuous random variable. It's gonna be a lot of fun! We'll use integration, we'll need to integrate by parts, and even use L'Hospital's Rule (L'Hôpital's Rule) to evaluate the limit of an indeterminate form!

The primary obstacle will be integrating a function of the form x*e^x from 0 to infinity. Skip to 2:25 to get straight to the integration if you don't need the introductory info on the exponential distribution and expected value of a continuous distribution.

If you are preparing for Probability Theory or in the midst of learning Probability Theory, you might be interested in the textbook I used in my Probability Theory course, called "A First Course in Probability Theory" by Sheldon Ross. 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!

I hope you find this video helpful, and be sure to ask any questions down in the comments!

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The outro music is by a favorite musician of mine named Vallow, who, upon my request, kindly gave me permission to use his music in my outros. I usually put my own music in the outros, but I love Vallow's music, and wanted to share it with those of you watching. Please check out all of his wonderful work.

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+WRATH OF MATH+

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[bobleonard]

All steps well explained and the general summary at the end is a nice touch. Keep up the good (and swanky) work sir!

[icecreamicequeen]

THANK YOU! I finally understand this! Thank you thank you!!

[CarlosCed]

I didn't understand any of it, but I watched the video because I enjoy that you can give an explanation to it all. 😅 I am not versed in math notation, but oh how wish I to learn math notation. It just makes life easier.

[palashkamble]

Very Helpful. Thank you so much. Subscribed right away.

[kulper]

would you please tell me the name of the whiteboard you used??

[asim_m]

Thank you sooo much!!!! Finally got it

[majdshamsedddine]

CRYSTAL CLEAR! LOVE THIS AND HERE IS A SUBSCRIPTION 🎉

[christophersheehy]

How do i find the expected value of 1 specific quantile? So E(8) for an exponential RV for example

[bhavyaanbarasan]

Hello sir, I am waiting for a deeper video on decomposition of graph, please sir