Moment Generating Function #2 (Continuous Distribution)

In this video I define the concept of Moment Generating Function and show how to derive it (using exponential density as an example). Exponential density function is not easy to work with however this will pay off as we review a lot of integration/exponential properties.

[lamees]

Your explanation made this difficult concept very easy. Thank you Thaaank youuu

[abnamibia]

This is actually what I was looking for. Thanks a lot. Stay blessed. I appreciate ya man.

[stepbil]

@betterkenya dude. Listen carefully to what i said in the vid. (t-λ) must be less then 0 in order for the integral to converge and not explode in our face. To convince urself let's assume that t-λ is a positive number say 2. And now evaluate this integral from 0 to infinity. U will notice that you cannot evaluate because the integral will blow out on the upper limit of integration (here infinity). hope it makes more sense now.

[fenekfenekk]

seek, and ye shall find. Thank you so much for making this video !!!

[apanapane]

Nice video. Thank you for taking the time to make this.

One question though. You say at around 6:50 that exp(x(t-λ) wouldn't converge unless (t-λ)<0.

However, if (t-λ) = 0, the expression would converge to 1. Does this provide a problem? Or is my reasoning faulty.

Cheers

[stepbil]

I think you have only focussed on the numerator. But what would happen to the denominator if we allow (t-λ) ≤ 0 and say let t-λ=0. We would end up with 0 in the denominator and we don't like that do we ;-)?

[apanapane]

Just realized that it wouldn't matter that exp(x(t-λ) would converge to 1 when (t-λ) = 0 since the next step anyway leads to e^0.

So the question is really, shouldn't the integral only be defined when (t-λ) ≤ 0 instead of <0 ?

[imkb]

Where did the expectation of e^tx come from? I realize that the MGF comes from it, but WHY? It is still mysterious and unintuitive to me why it works.

[stepbil]

You will find the key to this mystery if you watch #1 in the series.

[stepbil]

@Burrowsnakes . Well done Burrowsnakes! I thought I will wait an eternity for someone to spot it ;-).

[MrIlikecarrots]

If t-µ was positive it would mean you evaluate e^inf = inf. Then you would not be able to subtract the e^0(t-µ) value because you would have infinity - e^0(t-µ) which would still be infinity. To make this integral work you have to assume t-µ is negative so that you get e^inf(t-µ) = e^-inf = 0. Now you have zero to subtract your lower limit from.

[Omegamer]

I was happy that I got the right answer. And then 8:05 happened.

Damn you factorial... DAMN :P

[Tunnelsnakes]

typo at property #8: should be 5/5-3 not 5/5-2