# Probability distribution of Y in regression?

I'm trying to predict the probability distribution of $Y$ given $X_0, X_1, ...$ with a nonlinear regression. The probability distribution of $Y$ is likely not normal.

So far, I've set up and trained a regression model (support vector regression - SVR) which outputs the average value of $Y$, but I'm interested in seeing the entire distribution. What can I do?

(Please forgive me if this question doesn't make sense –– I'm very new to all this and still trying to figure things out. Any pointers would be appreciated!)

One thing you could do is look at quantile regression for different quantiles. That way you could for example predict how the $5^\mathrm{th}$ percentile differs for different values of your $X$s, how the $10^\mathrm{th}$ percentile differs for different values of your $X$s, etc.

If you build your method around semiparametric regression models you can estimate the entire distribution of Y for each X. A good example is the proportional odds ordinal logistic regression model. I have a case study doing this in my RMS book and course notes where at the end you'll see a nomogram displaying all sorts of predicted quantities (mean, quantiles, exceedance probabilities). See the chapter on ordinal models for continuous Y.