Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

If $X$ and $Y$ are two correlated Gaussian Random variables with parameters $(\mu_X, \sigma_X)$ and $(\mu_Y, \sigma_Y)$. If $Z=\exp(X - Y)$ then what is the distribution of $Z$.

Math world seems to suggest $X-Y$ should have a Normal Difference Distribution, based on this would that mean $Z$ has a log normal difference distribution ? I am unsure if there is any distribution named that. Also would the mean and standard deviation of $Z$ be simply $\exp(\mu(X - Y))$ and $\exp(\sigma(X - Y))$ where $\mu$ and $\sigma$ imply the mean and standard deviations one gets from Normal Difference distribution of $X-Y$.

Any help would be much appreciated.

share|improve this question

1 Answer 1

up vote 1 down vote accepted

If $X$ and $Y$ are joint normal, then $X-Y$ being a linear transformation of normal variables is again normal, implying that $\exp(X-Y)$ is log-normal. If no joint normality holds, log-normality may no longer follow. In general, you need to specify the joint distribution of $(X,Y)$ in order to derive that of $\exp(X-Y)$.

In case of joint normality, $X-Y$ is normal with mean, say, $\mu_d=\mu_X-\mu_Y$ and variance $\sigma_d^2=\sigma_X^2 + \sigma_Y^2-2 \text{Cov}(X,Y)$. In that case $\exp(X-Y)$ will a log-normal with those same parameters (that is, $\mu_d$ and $\sigma_d^2$). See this on how to specify the parameters of a log-normal distribution, and other information on that distribution.

Additionally, one point to be careful about here is that $\mu_d$ and $\sigma_d^2$ are the two parameters of log-normal distribution and NOT its mean and variance. For formulas for its two moments, you could look here.

share|improve this answer
    
Even if they are joint normal. What about a scenerio if both $X$ and $Y$ shared the same mean but difference standard deviations, let's say standard deviation of $Y$ is abotu $1/3$rd that of $X$. Then $X-Y$ would have a bimodal distribution ? would this still be a log-normal distrbution ? –  Hardy Dec 7 '12 at 5:49
    
It doesn't matter. I will edit my answer to clarify that point. –  Learner Dec 7 '12 at 5:52
    
Thanks i look forwrd to it. –  Hardy Dec 7 '12 at 5:54

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

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