Var (X) vs Var (X^2)

[ElizaSells]

Hi all - first post.

I was hoping for help with the following problem:

Let X be a discrete random variable and suppose Var(X)>1. How do Var(X) Var(X^2) compare?

One of the answers is "not enough information to solve" which is what I'm leaning towards...I can find plenty of examples where the Var(X^2) is bigger than the Var(X), but I feel like there is probably a counter example or an example where it depends on something. Can anyone help me? If you have a counter example could you explain how you thought of it? I can sometimes think of counter examples but I always feel as if Im just getting lucky. Thanks in advance.

 

[Dason]

Hint: Does it specify that the discrete distribution only takes on positive values?

 

[ElizaSells]

No, it doesn't specify. I tried doing it with negative values. I did the following:

X can take values from -3 to 2, with probabilities as follows (.2, .1, .4, .1, .1, .1)
I found the Variance of X to be 2.29 and the variance of X^2 to be 10.29. Am I missing something?

 

[Dason]

Try something a bit more simple. Think symmetric and only think about having two possible values.

 

[ElizaSells]

Ah! Ok I just did a RV that can only take 1 and -1, with a 50% probability of either value.

Var(x) is 1 and Var(X^2) is 0.

So it does depend, we can't always say the variance of X^2 is larger than the variance of X. Thank you so much!!

 

[ElizaSells]

Do you think this would change if it indicated the variance was less than one instead of greater than one? Im trying to determine if that had any effect on the problem...

 

[Dason]

Nope. Choosing -1 and 1 was only important in that they both have the same value when squared. You could use -.1 and .1 to the same effect. You can make var(X) as large or as small as you want and var(X^2) can still be 0

 

[ElizaSells]

Thanks Dason. I really appreciate your help.

 

[Muhammad Ayaz]

Hi all - first post.

I was hoping for help with the following problem:

Let X be a discrete random variable and suppose Var(X)>1. How do Var(X) Var(X^2) compare?

One of the answers is "not enough information to solve" which is what I'm leaning towards...I can find plenty of examples where the Var(X^2) is bigger than the Var(X), but I feel like there is probably a counter example or an example where it depends on something. Can anyone help me? If you have a counter example could you explain how you thought of it? I can sometimes think of counter examples but I always feel as if Im just getting lucky. Thanks in advance.

well. First of all its depend the intuition behind comparing X with X^2. Suppose if you want to check the nonlinear relationship between two variables, not only the value of x^2 will b greater but will the sign will be change. Check page # 724(15) for further understanding. Thanx