Strong vs Weak Law of Large Numbers

hello everyone today I want to share about my understanding about two concepts the strong law of large number and the weak law of large number which we have often we will often meet them in the introductory statistical course and this video serves for the purpose for you to explain the two concept to others without the need for rigorous mathematical proof so basically speaking these two concepts are derived under the same condition which I have written down here and they will raise to two different conditions which I will explain later so as for the conditions here we require at one as two or three this series of random variable to be iid that means identically independent independently distributed and we also require the expectation of the random variable to exist that means it cannot be positive or negative infinity and and then this come under this condition we have the following conclusion as shown here we let the expectation of the random variable as I should be mu and for the SN a simply means the sum of first n items so for the as n divided by n which is the number of observation is you can simply understand it at the sample mean well for the expectation of the random variable mu you can understand it as the population mean actually both both law will state that the sample mean will will converge to the population mean as the number of observation increase I think this is easily understood that the difference is here for the weak law for the weak law of large number it only required the convergence to be in probability and as for the strong law of large number it will require the convergence to be almost surely and this is a stronger version of the above one and I will illustrate it with the following graphs this graph is to illustrate the weak law of large numbers it is they I generated many realizations of the random variable as n over N and as you can see here as the number of observation and increase the realized value of SN over N which is the sample mean if you remember the value will converge their range of the value will narrow down and you can expect them to be become thinner and thinner and you can always draw such an interval here and you have the you have the confidence that such an interval can capture almost all the realized value of the sample mean however under the weak law of large numbers you cannot guarantee that all the that the probability of all the realization of the set of this random variable will definitely fall between this interval no matter how large you set the interval to be you cannot you cannot guarantee the probability of all the realized value to fall within this interval to be 1 however this is not the case for strong law of large number as you can see here this is just the 1 realization of the random variable as n over n that for any realization that this random variable we can always have an interval such that we can sure we can guarantee that the probability of the realized value of this of this random variable can follow the trend is interval with probability one and that this is why this version of law of Flash number is stronger than the previous one because in this case we can say it converge almost surely thank you for listening and we'll keep improving

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Id: Bn0wWZENeQI

Length: 4min 50sec (290 seconds)

Published: Sun Oct 29 2017