00:01
In this video, we are focused on finding the values of given probability based on chibb shift inequality.
00:07
So here we can say that that very inequality says that if there is a random variable x, such that it has a finite mean value, which is in our case is equals to basically 10, and a finite variance, which is in our case is equals to 4, then in that case the values of probability will be like we can say probability of absolute value of x minus mu and then here it is greater than equals to k sigma and this very expression is bounded between we can here right it will be less than equals to one divided by it will be here k square so it can be written like this now here we can say that there is another inequality and that can be written as absolute value of x minus mu and then it will be here less than we can say t value.
01:08
So this will be here greater than 1 minus sigma square divided by t squared.
01:16
Sometimes we write this very value of t as epsilon as well.
01:21
So, now here, from this very information, we have to firstly write that this is true for all t greater than 0.
01:28
And then here we can say that this is true for all here it can be written k value which is greater than 0.
01:35
So from here we can say that for the first part of this very problem, we have to find the value of probabilities such that here it can be written as absolute value of x minus mu and then it will be here we can say greater than.
01:51
Equals to 3 so from this very information we can say that this 3 can be written as probability of it will be here absolute value of x minus mu and then here it is going to be greater than equals to 3 multiplied with 2 divided by 2 so from this very information we can say that this will be less than equals to here 1 divided by 3 divided by two raise to the power two because here the value of sigma will be what the value of sigma will be two because sigma square is four right so from this very information we can say that we are going to have the value of this very expression which is probability of absolute value of x minus mu and then here it is going to be basically we can say how much mu value is 10 so it will be here greater than equals to 3 and that very value can be written that it is bounded with between 4 divided by 9.
02:54
So we can say that the answer for the first part of this very problem is coming around probability of modulus of x minus 10 and then here it is greater than equals to 3 and that will be here less than equals to we can say almost 0 .44.
03:12
So this is the answer for the first part of this very problem.
03:16
Now we have here another b part right.
03:19
So let us see how we can solve the b part.
03:22
So here in the b part we can write that the value of probability for here it will be x minus 10 and then here it is going to be lesser than 3 and that very value will be greater than 1 minus 4 divided by 9.
03:39
So this will be here equals to 5 divided by 9 and that comes out to be equals to here almost we can say 0 .56 .6.
03:49
And hence from this very information we can say that this is our answer for the second part so it can be put inside a box like this now here we have the c part in the c part we have to find the value of probability of 5 and then it is for we can say between basically x is between 5 and 50 so this will be here equals to p and then here we will be having 5 minus 10 and then here we will be having x minus mu so mu value is basically 10 and then here it will be basically 15 minus 10 and hence if we simplify this very expression we will be having the value such as let us here write it will be p and then here it is minus 5 less than x minus 10 this will be here less than 5 now from this very information we can say it will be probability of and then here it is absolute value of x minus 10 and then it will be here less than 5 divided by 2 multiplied with 2...