Lesson 15: Moment Generating Functions

welcome back everyone this is lecture number 15 and it's about moment generating functions or MGM's let's suppose that we have a random variable X with a support script X okay the moment generating function M G f of X is denoted by M subscript X of T and it is defined as follows the MGF of a random variable X is equal to the expected value of e to the TX okay that's the definition of MGF okay so this definition is true for discrete as well as continuous random variables so if X is discrete the MGF is simply the sum over all values of the support of X of e to the T X P of X of little X when X is discrete when X is continuous the only difference would be instead of summation we would have an integral over all the support of X of e to the T X times the PDF f of X DX that's when X is continuous F takes for occasion of e to the T X exists then we say the MGF exists for a random variable X Y is the expectation of e to the TX called the moment generating function recall that the first moment of random variable axis each of the expectation of X the second moment is expectation of x squared third moment is expectation of X to the power of three and so on and the cave's moment is expectation of X to the power of K so if we call em gf if we call em X of T a moment generating function we should be able to retrieve the moments if we are given the MGF but actually the the cave moment expectation of X to the K s the kids derivative with respect to T is the K of the moment generating function when we plug in a value of 0 for T all right let me explain that again so you have the moment generating function and you take the first derivative of the moment generating function so the first derivative with respect to t of the MGF of x after taking the first derivative plug in 0 for t and do you find that value is the expectation of X or the first moment of X loosely speaking I could write it in the following way and prime meaning the derivative of M with respect to T and I plug in 0 instead of T okay likewise the second moment will then be the expectation of x squared would be the derivative with respect to T remember this derivative is with respect to T twice when we plug in 0 that's loosely speaking and an appropriate way of writing it that would be the second derivative with respect to T of the moment generating function and then plug in 0 for T okay I will make another video proving that this is true okay I will provide a link for the proof of this note also that when you plug in 0 enter the MGF okay once the MGF MGF is the moment generating function is the expectation of e to the t x so if you plug in 0 into the MGF what happens is you're just finding the expectation of e to the 0 times X so this is the expectation of 1 this is the constant so that gives you 1 so when you plug in 0 into the MGF your value the value that you find is going to be equal to 1 let's do an example let's do an example alright let's say we have a discrete random variable X with the following PMF probability mass function so the PMF is given by 3/4 times 1 over 4 to the power of X and X could take values of 0 1 2 3 all the way up to infinity so let's find the MGF the MGF of x MX of T is equal to the expectation of e to the T of X which is since this is a discrete random variable that's going to be the sum from the support is from 0 to infinity discrete values from 0 to infinity of e to the T X times the PMF which is 3 over 4 times 1 over 4 to the power of X okay now I can take out the constant outside of the summation 3 over 4 times the summation X from 0 to infinity remember here e to the T X is the same as e to the T to the power of X so I can combine this this is 1 over 4 to the power of X this is e to the T X which is e to the T to the power of X combining the two I would get e to the T divided by 4 to the power of X now you if you look carefully here this is a geometric series with a common ratio of e to the T divided by 4 remember from your calculus when you have a geometric series summing from 0 to infinity of RR let's say R is a common ratio to the power of X ok this is equal to R to the power of 0 which is 1 plus R to the power of 1 plus R squared plus R to the power of 3 plus and so on this sums to 1 over 1 minus R when R is less than 1 ok whenever R is less than 1 that summation is equal to 1 over 1 minus R you you you must remember this this is a very very crucial identity and you would need this and the following lectures to come and and you would need this and an example you definitely need this on example so if e to the T divided by 4 is less than 1 so if whenever e to the T divided by 4 is less than 1 then this is going to be equal to 3 divided by 4 times 1 over 1 minus e to the T divided by 4 this implies the MGF of X is 3 over 4 divided by 1 minus e to the T divided by 4 whenever a to the T divided by 4 is less than 1 by the way which forms an up and neighborhood around 0 so by definition MGF is valid if the expectation of e to the T X exists in an open neighborhood around zero what is the expected value of X if we know the MGF one way to find this by the way you could have you could have decided to find the expectation in the following way expectation as the summation from 0 to infinity of x times 3 over 4 times 1 over 4 to the power of X it turns out this way is actually much more difficult than this way using the MGF method okay so if I know the MGF finding it was not very difficult if I know the MGF what I have to do is I just need to find the derivative of the MGF with respect to T and then plug in 0 for T ok that gives me the expectation of X so what is the derivative of the MGF with respect to T that's the derivative of this quantity and using power rule and the chain rule you would find out that this is negative 3 over 4 divided by 1 minus e lemmy 1 minus e to the power of T divided by 4 square then utilizing the chain rule I need to multiply this by negative e to the T over 4 that's the derivative of the inside here ok so that is equal to 3 over 4 times e to the T divided by 4 times 1 minus e to the T over 4 squared so when I plug in 0 I get the expectation so the expectation of X as we said is the derivative of the moment generating function evaluated at 0 so plug in 0 for T so that's going to be plug in 0 on e to the t you'd find e to the power of 0 is 1 here also e to the power of 0 is 1 so I would have 3/2 by 4 over 4 times 1 minus 1 over 4 square and simplify this you would get a value equal to 1/3 so the expectation of X as 1/3 and likewise you could find the second moment the third moment and so on before I do one more example I would like to note the following note that the MGF can also be written in the following way MGF is the expectation of e to the T X ok so if I do a Taylor series expansion of e to the T X if I have a Taylor series expansion about 0 of e to the TX I would find the following that would be T times X to the power of 0 T times X to the power of 0 which is 1 plus T times X to the power of 1 divided by 1 which is T times X plus T times X square divided by 2 factorial plus T times X cubed divided by 3 factorial and so on it's an infinite series ok so this is equal to the expectation of 1 which is 1 plus the expectation of T times X as T times the expectation of X plus now if you if you look at this I can write that as T squared divided by 2 factorial times x squared but T squared over 2 factorial is a constant but X square is random so I would need to put the expectation on it ok likewise you see this is going to be T cubed divided by 3 factorial times e to the power of x cubed plus and so on so the another way to write the MGF is in the following way that's going to be on the sum K equals 0 to infinity of T to the power of K times the expectation of X to the K which is the cave moment divided by K factorial K factorial so the SOA may give you an example like following they may actually just give you an mg of a random variable and they tell you it's 1 plus T times some number plus T squared divided by 2 factorial of some number plus and so on so what you need to do is simply figure out oh this is mg F written in the following way so I can reconstruct it to find the moments then you'd find the moment and you will be ant you'll be able to answer the questions one last example the value of a piece of factory equipment after three years of years as one hundred times 0.5 to the power of X where X is a random variable having a moment generating function provided here okay calculate the expected value of this piece of equipment after three years of years so if I let Y to be equal to 100 times 0.5 to the power of X the question is to find the expected value of y which is the expected value of 100 times 0.5 to the power of X so since 100 is a constant I can take it outside of the expectation and I have the expectation of 0.5 to the power of X since I know the moment generating function of X I can rewrite this expectation in the following way so this is 100 times the expectation of e the natural exponent the log like I should probably use the natural log of 0.5 to the power of X this is by property of exponents okay this is equal to this expression is equal to 0.5 to the power of X so this is equal to 100 times the expectation of by property of logs what we know is we can take the power to be the coefficient so that's going to be e times x times the natural log of 0.5 okay now you have the MGF evaluated at X at E sorry the MGF valuated at t equals the natural log of 0.5 recall mg f of x is the expectation of e to the t x or x times t does matter so I have X I have T here so this is T the value of T you are given the MGF what you need to do is simply plug in the value the natural log of 0.5 and 2t here so this is going to be equal to the MGF of X evaluated at the natural log of 0.5 and that equals 100 times 1 over 1 minus 2 times log the natural log of 0.5 put that in your calculator and you will find a value equal to 41 point 9 0 60 okay thanks for watching this lecture

Channel: Stat Courses

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Keywords: Probability, MGF, Statistics, Actuarial, Actuarialpath, Actuarial Science, Moment Generating Functions, Actuarial Exams, Society of Actuaries Exam P, MGFs, probability and statistics, calculate expected value using MGFs, calculate expected value using moment generating functions, first moment and moment generating functions, moments and moment generating functions, calculate variance using MGF, calculate variance using moment generating function

Id: cbmfYoepHPk

Length: 16min 53sec (1013 seconds)

Published: Wed Feb 13 2013