What is the variance of X/Y? Does it exist?

  • Thread starter mqwit
  • Start date
  • Tags
    Variance
In summary, the problem with finding the solution to Var(X/Y) is that the moments of 1/X may fail to exist without more information about the probability distributions of X and Y. This is because the denominator could potentially make the moments of 1/X non-existent. To find the solution, the joint distribution of X and Y would also need to be specified.
  • #1
mqwit
1
0
I know Var(aX + bY)...
I know Var(XY)...
I don't know of a solution to Var(X/Y)... is there one?
 
Physics news on Phys.org
  • #2
Not a quick one. In fact, I'm not sure what "solution" to [tex] Var(XY) [/tex] you mean, unless the two variables are (stochastically) independent.

The problem is with the denominator - it is entirely possible for a random variable [tex] X [/tex] to have finite mean, variance (even higher-order moments) but for the moments of [tex] 1/X [/tex] fail to exist. Without more information there is no other answer to provide.
 
  • #3
You don't mean X divided by Y do you? You mean X given Y?

- Warren
 
  • #4
I don't see how you can answer any of those questions without specifying the probability distributions of X and Y.

If, for example, X is 1 with probabilty 1/2 and 2 with probability 1/2, Y has the same distribution, then X/Y is 1/2 with probabililty 1/4, 1 with probability 1/2, 2 with probability 1/4 so the mean is (1/4)(1/2)+ (1/2)(1)+ (1/4)(2)= 9/8 and the variance is (1/4)(1/4- 9/8)2+ (1/2)(1- 9/8)2+ (1/4)(2- 9/8)2= (1/4)(-7/8)2+ (1/2)(-1/8)2+ (1/4)(7/8)= (1/4)(49/64)+ (1/2)(1/64)+ (1/4)(49/64)= 100/256= 25/64.

Now, what probability distribution are you talking about?

(And if "X/Y" means "X given Y" rather than "X divided by Y" you will need to specify the joint distribution as well.)
 
Last edited by a moderator:

1. What is the variance of X/Y?

The variance of X/Y is a measure of the spread or variability of the ratio of two random variables X and Y. It is calculated by taking the square of the standard deviation of X/Y, which in turn is the square root of the variance of X/Y.

2. How is the variance of X/Y calculated?

To calculate the variance of X/Y, you first need to calculate the standard deviation of X/Y by taking the square root of the variance of X/Y. The variance of X/Y is then calculated by taking the square of the standard deviation of X/Y.

3. Does the variance of X/Y always exist?

No, the variance of X/Y does not always exist. It depends on the values of X and Y and their relationship with each other. If Y is equal to 0, then the variance of X/Y does not exist as division by 0 is undefined.

4. What does a high variance of X/Y indicate?

A high variance of X/Y indicates that there is a large spread or variability in the ratio of X and Y. This means that the values of X and Y are not close to each other and may be highly different from each other.

5. How is the variance of X/Y useful in scientific research?

The variance of X/Y is useful in scientific research as it helps to understand the relationship between two random variables X and Y. It can also provide insights into the variability of the ratio of X and Y and can be used to make predictions or draw conclusions about the data being studied.

Similar threads

  • Calculus and Beyond Homework Help
Replies
8
Views
1K
  • Calculus and Beyond Homework Help
Replies
9
Views
2K
Replies
0
Views
678
  • Calculus and Beyond Homework Help
Replies
2
Views
1K
  • Calculus and Beyond Homework Help
Replies
14
Views
1K
  • Programming and Computer Science
Replies
1
Views
415
  • Calculus and Beyond Homework Help
Replies
3
Views
2K
  • Calculus and Beyond Homework Help
Replies
2
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
5K
  • Calculus and Beyond Homework Help
Replies
3
Views
1K
Back
Top