Can E(Y/S) Be Less Than 1 in IID Random Variables?

[kiwikiwi79]

Mathematical Statistics~help please!



Please help me with these question...

Let S and Y be i.i.d positive random variables with E(X)<infinity.
Show that E(Y/S) is greater or equal to 1. Give an example where

E(Y/X)=positive infinitiy...

I have no Idea where to begin...please help me..

 

[matt grime]

E(Y/S) = E(Y)(E1/S) since they are independent.What does it now boil down to showing? What results do you know that tell you when E(f(X)) > f(E(X)) for some function f?

 

[tacman]

Hi there! Mathematical statistics can definitely be tricky, so I understand your frustration. Let's break down the problem step by step.

First, let's define what it means for two random variables to be i.i.d. This stands for "independent and identically distributed," meaning that the variables have the same probability distribution and are independent of each other. In this case, S and Y have the same probability distribution and are independent.

Next, we need to use the fact that E(X) is finite. This means that the expected value of X exists and is not infinite. We can use this fact to help us solve the problem.

Now, let's look at the expression E(Y/S). This is the expected value of Y divided by S. We want to show that this is greater or equal to 1. To do this, we can use the fact that S and Y are i.i.d. This means that we can rewrite the expression as E(Y/Y), since S and Y have the same probability distribution. Using the definition of expected value, we can rewrite this as the integral of Y times the probability density function of Y, divided by the integral of Y.

Since we know that E(X) is finite, we can use this to show that the integral of Y is also finite. This means that the integral of Y/Y is also finite. And since the integral of Y/Y is just 1, we can conclude that E(Y/S) is greater or equal to 1.

For the second part of the problem, we need to find an example where E(Y/X) is positive infinity. This means that the expected value of Y, when divided by X, will result in a positive infinity. One example of this is when X is equal to 0. In this case, any value of Y will result in a positive infinity when divided by 0. So, we can say that E(Y/X) is positive infinity when X is equal to 0.

I hope this helps! Remember, when solving mathematical statistics problems, it's important to break down the problem step by step and use the definitions and properties of the variables to guide your reasoning. Good luck!