4*exp(-4x) is the density of an exponential random variable X with lambda=4, so the expected value of X is pretty much exactly this integral, except for a factor of 4, so this integral is 1/4*E(X) and E(X) = 1/lambda = 1/4, so we get 1/16
Then again, relying on the expected value of a random variable is not a reliable method to solve this type of problem. If you didn’t know the expected value of an exponential random variable, you’d still have to solve the integral, most likely in your way
@@NumberNinjaDaveDoesn’t IBP itself involve a formula? Seems like the best course is to select the formula that requires the least effort. If the goal was simply to demonstrate IBP, the video succeeds. Keep up the good work. 😊
@@chrisjuravich3398 I’m glad the video got your “stamp of approval” 😉 did you actually watch the whole video? Let me know if it suggested understanding of IBP The point of my videos are to teach understanding and not plug and chug formulas. But I also pride myself on not dragging on for hours to teach the concepts Understandably, as the channel grows, some will question the efficacy of the content and so I look at the overall trajectory of the channel growth for what I determine as what defines the content as “succeeding”.
At 5:45 were you supposed to make that x/e^4x instead of x/e^-4x?
Yup, you caught the error! Nice catch! Instead of redoing the film, I rolled with it as a fun exercise, haha
4*exp(-4x) is the density of an exponential random variable X with lambda=4, so the expected value of X is pretty much exactly this integral, except for a factor of 4, so this integral is 1/4*E(X) and E(X) = 1/lambda = 1/4, so we get 1/16
Thanks for chiming in, ninja 🥋
Then again, relying on the expected value of a random variable is not a reliable method to solve this type of problem. If you didn’t know the expected value of an exponential random variable, you’d still have to solve the integral, most likely in your way
I just became a ninja today 😊
Best comment of the day. Terry Silver welcomes you
gamma function kills it very fast...
Of course, although, this video is useful for those who want to understand integration techniques versus plugging into a formula
@@NumberNinjaDaveDoesn’t IBP itself involve a formula? Seems like the best course is to select the formula that requires the least effort. If the goal was simply to demonstrate IBP, the video succeeds. Keep up the good work. 😊
@@chrisjuravich3398 I’m glad the video got your “stamp of approval” 😉 did you actually watch the whole video? Let me know if it suggested understanding of IBP
The point of my videos are to teach understanding and not plug and chug formulas. But I also pride myself on not dragging on for hours to teach the concepts
Understandably, as the channel grows, some will question the efficacy of the content and so I look at the overall trajectory of the channel growth for what I determine as what defines the content as “succeeding”.
Subscribed!
And, yes, I watched the whole video. 😊
@@chrisjuravich3398 I appreciate you, ninja 🥷
QUIZ: Can you simply plug in "infinity" when evaluating the integral? Why or why not
Do you have an alternative way of solving this integral from how I did it? Let's hear what you did!
Gamma function.
U-sub for 4x and simplify. You will get 1/16 * Γ(2) which equals to 1/16
just use the DI table method, it ends in a matter of seconds
What if a student forgets the formula on an exam? I teach understanding and not memorization