Moments and Moment Generating Functions - Expectations and MGF calculation in Tamil.

[Music] x into variable value into its probability uh [Music] x variance of x is equal to e of x minus e of x the whole square in a molecular a x exporter square evaluating square root of also common number of variance of a x equal to a square into variance of x and the formula applies no problem resulting next moments about origin expectations expectation of x minus of x power r the whole power x power or dash not dash i think mu r mu r is moments of moment about mean okay moment about r in a symbol mu or dash moment about the origin okay the moment so are about about origin x zero because origin is always zero upon zero x minus zero and simple x power r is moment about the origin e of x minus x bar the whole power r is moment about the mean okay x minus x bar for example mu one negative mu one negative number o e of x minus x bar the whole power one row that is equal to okay so that is that is x bar itself expectation of a constant is equal to constant itself okay expectation of x naught constant that is x bar upper overall inertia x bar minus x bar that is equal to zero so so mu one equal to zero so irrespective of the distribution irrespective of the probability density function or distribution function if predominantly in the mu one not a value april always mu 2 is equal to e of x minus x bar the whole square in square minus 2 x x bar minus x bar the whole square that is equal to individual terms i will depend upon my expectation that is a of x square minus 2 constant up here because the x borrow that is equal to your x square minus 2 into x bar e of x bar is again x bar and x bar is a constant minus a p then now x bar square x bar and x bar equal to x bar square 2 x bar square minus 1 x bar square that is equal to your plus circle that is equal to e of x square minus x bar square number x bar square n x bar square is nothing but e of x the whole square what is this this is nothing but variance of x this is nothing but variance of x so a poor number mu two number calculate under the another central moment about the uh central moment of order two another variance moments about [Music] generating function of the variable x and it is a function in terms of t or new variables generating function in the variable order moment generating functions okay so this is the expansion and formula for moment generating function okay so mx of t equal to e of e power tx formula very good i'm a problem next calculation foreign differentiation okay so differentiate panel function after the differentiation differentiation put t is equal [Music] p of x is equal to x equal to one by six comma x is equal to one two three four five six in the problem manually uh or division outcomes in a possible outcomes in a one two three four five six oh one kind of probability number equal okay over daily one critical kind of probability over uh daily one kind of probability one basics two kind of probability one basically equal divided it's an unbiased die so probability of x is equal to x equal to [Music] [Music] first value is one so one t into its corresponding probability though the corresponding probability in a one by six the okay plus e of e of x i is equal to 2 of dinner e power 2 t into its corresponding probability 1 by 6 e power 3 t into its corresponding probability 1 by 6 so 1 by 6 into e power 4 t plus 1 by 6 into e power phi t plus 1 by 6 into e power 6 t that is equal to 1 by 6 a common addition n number of e power t plus e power 2 t plus e power 3 t e power 4 t plus e power is equal to m x t or the order so r derivative and then t is equal to zero you putting up watching one note two or three o in a minimum differential right if but now let's pray a particular into over a term by term differentiation e probability differentiation in our e power t itself e power t itself plus e power 2 t differentiate potential 2 into e power 2 [Music] differentiation 3 into e power 3 d plus e power 4 differentiation 4 into e power 4 t plus 5 into e power 5 t plus 6 into e power 60 is equal to 0 0 0 0 0 the e power t one plus two into e power t so one plus three into e power three t so one plus four a day mother plus five plus six okay so in the simple way for now i'm gonna tell tx into continuous random variable the probability function that is probability density function is f of x integral d x minus infinity to plus infinity that is equal to right formula e power minus x by 3 dx that is equal to okay constant required constant so it is 1 by 3 integral 0 to infinity render our product of e into t power x into e power minus x by 3 so t x minus x by 3 d x up divided by okay so product of length exponential multiplied by motion divided by infinity uh is equal to zero something anything divided by infinity is zero i don't so automatically infinity substitute integrate point number o e power exponential of that thing that is equal to e power minus of 1 by 3 minus t into x divided by minus of 1 by 3 minus t limits from 0 to infinity into upper one by three minus t or minus zero so minus in a velocity uh one by three minus t and i'm looking at the number of one minus three t divided by three zero so one minus three t divided by three even the lower will become zero that is e power minus zero that is equal to uh so this is equal to in even the value and now zero in the value anything power zero is one in the value one now go so number in the bracket in the valence terminal minus one minus one with level and the minus one when the automatic unit of the plus one i don't know so opening a simplified pending abdina one divided by 1 minus 3 t this is your moment generating function of the given uh probability density function f of x it's downloaded generating function simple numbers the minus one to send the positive i don't know so finally one divided by one minus three t is the moment generating function of the given uh variable x okay so moment generating functions um formula ponyta object that is 1 minus 3t the whole power minus 1 after the differentiation put t is equal to 0 you put differentiate from llama 1 minus 3t the whole power minus 1 other single variable a single variable so minus 1 into 1 minus 3t minus 1 minus 1 that is equal to minus 2 into bracket let's get the momentum our body differentiate that is minus 3 that is equal to um 3 into 1 minus 3 t the whole power minus 2 now put t is equal to 0 t is equal to 0 available to 0 and i know 1 minus 0 minus three one minus three t the whole power minus [Applause] f of x dx limits from 0 to infinity given then the problem is 0 to infinity that is equal to integral from 0 to infinity x into f of x energy 1 by 3 e power minus x so actually it is a shortcut okay next e of x square calculate e of x square formula in mx of t double dash then t is equal to 0 that is equal that is 3 into minus 2 into 1 minus 3t the whole power minus 3 into minus 3 again at the cupra t is equal to 0 that is equal to upper minus six minus is minus into minus plus six into nine simplified nobody never is it three into minus two so minus six minus six into minus three equal to plus eighteen so answer is 18 e of x square but you know what i'm a cancer verb indicator so variances of x square minus z of x the whole square that is 18 minus 3 square is 9 18 minus 9 will give you 9 so variance of x is 9 and the mean of x is 3 uh so actually generating function i'm a use partner uh [Music] b you

Channel: MATH THE IMMORTAL - என்றும் அழியா கணிதம்

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Length: 42min 2sec (2522 seconds)

Published: Sat Jul 17 2021