I am given a bivariate distribution for X and Y and then the marginal distribution for Y. The only metrics I know for these are their mean and variances (Expected value and Variance of bivariate distribution and again for the marginal). But I also have a drawn sample from each distribution. I ultimately want to find the variance of the marginal distribution for X and I know that the marginal mean of X equals the mean of the bivariate distribution of X and Y. Each distribution sample has 2 numbers. So each record consists of a pair of an actual number and an expected number (e.g. 25 samples of actual and expected numbers from the bivariate distribution and 25 samples of actual and expected numbers from the marginal Y). So therefore you can calculate an error for each sample record. Given this information, I want to come up with a best way to estimate the variance of X. Can someone advise?
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$\begingroup$ Do you know if $X$ and $Y$ are independent random variables? $\endgroup$– Matt BremsNov 14, 2015 at 22:58
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$\begingroup$ X and Y are definitely not independent. $\endgroup$– BulaunNov 15, 2015 at 22:54
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$\begingroup$ Do you have the functions $f(Y)$ and $f(X,Y)$? $\endgroup$– Matt BremsNov 15, 2015 at 22:57
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$\begingroup$ I don't know the entire distribution for Y and (X,Y). I just know their means and variances. Further we have a drawn sample from each of the Y and (X,Y) distributions and for each draw I get an expected drawn number and the actual drawn number. Only other information I know is that the mean of (X,Y) = mean of X. $\endgroup$– BulaunNov 15, 2015 at 23:10
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$\begingroup$ I understand I can calculate sample standard deviations for the Y and (X,Y) distribution. How can I arrive to an estimate of the variance of X with this information? $\endgroup$– BulaunNov 15, 2015 at 23:21
1 Answer
You cannot sample an observation from $f(X,Y)$ and get one number. You sample an observation from $f(X,Y)$ and you should get something like $(x_1,y_1)$. As such, you should have a list of data points from $X$.
I would suggest calculating the sample standard deviation of your $X$ data given that you know $E[X]$ and have a sample of data points from $X$. You can calculate $s_X$ and then note that $s^2_X$ is the sample variance of $X$.
It is also an identity that $Var[X]=E[X^2]-E[X]^2$. You know what $E[X]$ is from the problem and you can empirically estimate $E[X^2]$ by squaring all of your $X$ data and calculating the sample mean of that.
These are two methods that should get you to estimate the variance. If they seem pretty close, then you're probably in the ballpark.
What my other thought would be is that, analytically, $f(X)=\int_{-\infty}^{\infty}\frac{f(x,y)}{f(y)}dy$. I don't know how you could analyze this without knowing your functions $f$ and your sample size seems too small to estimate density functions.
