1
$\begingroup$

Suppose $X_{11}, ...X_{1n}$ are iid rvs following an exponential distribution with expectation $\mu_1$, and $X_{21}, ...X_{2n}$ are iid rvs following an exponential distribution with expectation $\mu_2$. All rvs are mutually independent.

Let $A=\frac{1}{n}\cdot(X_{11}+...+X_{1n})$ and $B=\frac{1}{n}\cdot(X_{21}+...+X_{2n})$

For large $n$, what is the normal approximation to the distribution of $A/B$?

Any ideas would be greatly appreciated!

$\endgroup$
4
  • $\begingroup$ Why would you need a normal approximation? You can compute the distribution exactly! (In any case, first you'd need to know that it does have a suitable normal approximation -- the hard part is showing that and appropriately standardized ratio is asymptotically normal, though if you can rely on a couple of theorems that should be easy enough. The really easy part is figuring out the parameters, since once you know the distribution of the ratio you can look those up.) $\endgroup$
    – Glen_b
    Oct 1, 2015 at 1:45
  • $\begingroup$ Welcome to Cross Validated! Please add the self-study tag, read its tag-wiki and modify your question to follow the guidelines on asking such questions. In particular, you'll need to clearly identify what you've done to solve the problem yourself, and indicate the specific help you need at the point you struck difficulty. $\endgroup$
    – Glen_b
    Oct 1, 2015 at 1:49
  • $\begingroup$ Do you know what theorems should be used to derive the distribution of the ratio? $\endgroup$
    – user90890
    Oct 1, 2015 at 2:42
  • $\begingroup$ There might be one way to derive the exact distribution, but the probem i got is ti find the normal approximation when n gets large. I thought that I can write A/B as 1/n*[(X11/(X21+...+X2n)+•••+(X1n/(X21+...+X2n)]. If i can find the common distributiob of each element in the bracket, then i will be able to find the normal approximation of A/B. The numerator (X11) follows exponential distribution, while the denominator follows a gamma distribution. But i am not sure what the distribution it is for the ratio of an exponential and gamma. $\endgroup$
    – user90890
    Oct 1, 2015 at 2:53

1 Answer 1

1
$\begingroup$

The easiest way is still to calculate the exact distribution first.

  1. a. Calculate the distribution of the numerator.
    b. Calculate the distribution of the denominator.
    c. Use a result you'll already know about the ratio of independent variates from a distribution obtainable by a simple rescaling of those.
    d. Now, what's the mean and variance of the distribution of the ratio?

  2. Also, take a look at Slutsky's theorem; if you set things up right, you may be able to avoid worrying about the denominator except as a scaling factor. NB don't just deal with the raw ratio, which converges to a degenerate distribution.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.