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

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The discussion centers on the variance of the ratio of two random variables, specifically Var(X/Y). It is established that without knowing the probability distributions of X and Y, it is impossible to determine Var(X/Y). The conversation highlights that while Var(aX + bY) and Var(XY) can be computed under certain conditions, Var(X/Y) poses unique challenges due to the potential non-existence of moments for 1/X. An example is provided where X and Y are identically distributed, leading to a calculated variance of 25/64 for the ratio X/Y.

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I know Var(aX + bY)...
I know Var(XY)...
I don't know of a solution to Var(X/Y)... is there one?
 
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Not a quick one. In fact, I'm not sure what "solution" to Var(XY) you mean, unless the two variables are (stochastically) independent.

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

- Warren
 
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 probability 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.)
 
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