# How can I calculate the probability that the product of two independent random variables does not exceed $L$?

I have one variable, $X$, which is provided hourly for a period of one month (720 total values in the series). I have another variable, $Y$, which is provided quarterly (for which I am provided the average value and standard deviation, but not sure how many values in the population).

I am trying to find out the probability that the product of these values ($Z=XY$) will not exceed some limit, $L$.

Question: How can I calculate the standard deviation (thus the probability of exceeding $L$) of this product, $Z$, if the number of measurements is not equal in $X$ and $Y$?

• $X$ and $Y$ are random variables, and you are interested in the random variable $Z$ that is the product $XY$. Assuming that $X$ and $Y$ are independent, the variance of $Z$ can be calculated following this thread: stats.stackexchange.com/questions/52646/…, and the variances of $X$ and $Y$ can be estimated using the sample variance. If the random variables are not independent, then you may be screwed as you need to be able to pair the observations in order to estimate the covariance.
– Alex
Sep 7, 2016 at 5:45