Question 4 the real number m is called a median of the...

The real number m is called a median of the distribution function F whenever lim F(y) ≤ F(m) for all y < m and lim F(y) ≥ F(m) for all y > m. Show that every random variable has at least one median.

Transcript

00:01 Hello friends, today we are going to solve this problem that says we have to show that each distribution function f has at least one median and that the set of medians of f is a close interval of r.

00:15 Now let us try to solve this problem.

00:17 First of all, i can say that assume y to be a random variable, sorry.

00:22 Now i be a closed interval.

00:25 Okay.

00:25 So first of all, i can say that we have been given that each distribution.

00:30 Function n has to have at least one medium right so with respect to that very statement we are going to assume our first case as if p p or y belongs to i equals half at least one median of y is in i at least one median of y is in i so this statement clearly shows us that very thing because we have been given one more thing in the problem right and what is it it is let me write it here it is given that limit f of y and then it will be uh less than equals to half and then it will be less than equals to f of n so from here what you can you know confer that uh let me you know write this very thing that from this part you can just confer whatever i'm going to write from here so if you read out this of second case it says consider i has fallen property if i is any is any closed interval such that i is a proper subset of i then p of j is greater than how okay so let us see how this problem can be solved further i can write from here that so i contains all medians of oil so what i contains all medians of f so from here i can say that assume f equals distribution function of y then what will happen i can just say that if i equals minus infinity to b and then you must remember that i have taken uh not closed interval okay from here till now so then f of b will be equals to how this is an assumption hence b is median of x i hope you are able to understand this very statement that as b is median of x and how is it it is because that if i is belonging to minus infinity to b, then f of b equals to a half, right? so from that half, i can say that this is an assumption, hence, p is a median of x.

02:41 Now, if i belongs to a to infinity, then f of a will also be equals to half.

02:46 So, a is also a median of x.

02:48 Now, from these two of cases, i can say that if a belongs to a to b, then f of a is less than equals to half and f of b is greater than equal to half...

Question

The real number m is called a median of the distribution function F whenever lim F(y) ≤ F(m) for all y < m and lim F(y) ≥ F(m) for all y > m. Show that every random variable has at least one median.

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