# Distribution of function of variable having a Gaussian distribution

If I have a variable $X$ whose Gaussian distribution is known and let $f$ be a known function, is there a way to compute the distribution of $f(X)$ i.e. the resulting Gaussian distribution from this?

Is the result actually a Gaussian distribution?

• $X\sim\mathcal N(\mu=0,~\sigma^2=1),~~f(X)=2$ – gung May 9 '13 at 2:25
• @gung Technically in your example $f(X)$ does have a Gaussian distribution. But the almost as simple function $X^2$ clearly is not Gaussian, because it can never be negative. Sree Prasanna, what exactly do you mean by "compute" $f(X)$? It can always be expressed as an integral--is that a "computation"? Or do you mean being able to give a closed-form formula whenever $f$ itself has a closed formula? Or do you mean just being able to compute probabilities numerically to any reasonable desired precision? Or something else? – whuber May 9 '13 at 3:31
• @whuber Oh, my apologies, I didn't see you'd already done the same case in comments as I just did. I'll do another one as well, to make up for it. – Glen_b May 9 '13 at 7:04
• @Dilip The "only when" is not correct. For instance, let $A$ be a Lebesgue measurable proper subset of positive reals and define $f(x)=-x$ whenever either $x \in A$ or $-x \in A$ and otherwise let $f(x)=x$. This $f$ does not have the form you propose, yet when $X$ has a Normal$(0,\sigma^2)$ distribution, so does $f(X)$. – whuber May 9 '13 at 14:31
• @whuber Thanks for pointing out the erroneous statement. I remember the counterexample but it was not at the forefront of my mind. I am deleting my previous comment and replacing it with a corrected one. – Dilip Sarwate May 9 '13 at 17:58

I think you are talking about change of variables.

If X is a continuous r.v. with pdf $f_X(x)$ and sample space S, if $g: S \rightarrow T$ is an invertible transformation with differentiable inverse $h = g^{-1}$, and $Y = g(X)$, then Y is a continuous r.v. with pdf $f_Y(y)$ defined by

$f_Y(y) = f_X(h(y)) \cdot |h'(y)|$.

For example, suppose $X \sim exp(\lambda)$, what is the distribution of $X^2$?

Well, here we have $g: (0,\infty) \rightarrow (0,\infty)$ defined by $g(x) = x^2$. The inverse if $h(y) = g^{-1}(y) = \sqrt y$

Then $h'(y) = 0.5 y^{-1/2}$.

The density of X is $f_X(x) = \lambda e^{-\lambda x}$ so $f_X(h(y)) = \lambda e^{-\lambda \sqrt y}$.

And finally multiplying by the derivative gives $$f_Y(y) = f_X(h(y)) \cdot |h'(y)| = \lambda e^{-\lambda \sqrt y} \cdot 0.5 y^{-1/2}$$

This simplifies to $$f_Y(y) = \frac{\lambda e^{-\lambda \sqrt y}}{2\sqrt y}$$

There is your density for a function of a random variable.

• Yeah, that's the actual way to do it. I understood that after reading that in a book. Thanks. – SPRajagopal May 14 '13 at 9:28
• @LindaH +1 Well constructed answer; it seems like perhaps you had more insight into the OP's underlying need than I did. – Glen_b May 14 '13 at 12:44
• @Glen_b In that case, stats.stackexchange.com/questions/14483/… is extremely closely related. – whuber May 14 '13 at 17:47

Addressing the last question in particular --

1) Consider $X$ being standard Gaussian (mean 0, variance 1), and $U = \Phi(X)$ where $\Phi()$ is the standard normal cdf. Does $U$ have a Gaussian distribution? Let's simulate (in R in this case):

 u <- pnorm(rnorm(100000L))
hist(u,n=300)


... nope.

In fact you can work out that it must be standard uniform.

2) Consider $X$ being standard Gaussian (mean 0, variance 1), and $Y = f(X) = X^2$.

Does $Y$ have a Gaussian distribution? Let's simulate (in R):

 y <- rnorm(100000L)^2
hist(y,n=300)


and what do we see?

... nope.

In fact you can work out that it's chi-squared(1).

In particular circumstances, certain results imply you can obtain an approximately Gaussian distribution... but it's not the general case. For example, if the mean is many standard deviations from 0, (e.g. $X\sim N(5,0.1^2)$), then $Y=X^2$ is at least approximately normal:

 y <- rnorm(100000L,5,.1)^2
hist(y,n=300)


... and similarly $\exp(X)$, and $X^{1/3}$ and $log(X^2+\sqrt\pi)$, and a whole menagerie of other transformations will also give approximately Gaussian distributions in this case.

Edit:

You do get exact normality if you transform a Gaussian random variable with a linear transformation, so, as @DilipSarwate points out, if $X$ is Gaussian, $Y = a+bX$ is Gaussian (the case $b=0$ which was discussed by @gung and @whuber in comments is 'degenerate', but usually still counted as Gaussian when considering the whole location-scale family of Gaussians).

It's not quite the case that you can get to a normal only by linear transformation, though if you are restricted to monotonic transformations, I think this will be the case.

A nonlinear transformation of a normal that yields a normal:

Let $X$ be standard Gaussian. Let $F_1$ be the cdf of a $\chi^2_1$ random variable and $\Phi$ be the cdf of a standard normal. Then $\Phi^{-1}(F_1(X^2))$ is a nonlinear (and non-monotonic) transformation of $X$ where the resulting random variable is Gaussian.

y <- qnorm(pchisq(rnorm(100000L)^2,1))
hist(y,n=200)


In respect of the first question, if $Y=h(X)$ you can in principle work out the distribution of $Y$. If $X$ is continuous and $h$ is invertible, it goes like this:

If $F_X$ is the cdf of $X$ then $P(Y\leq y) = P(h(X)\leq y) = P(X\leq h^{-1}(y)) = F_X(h^{-1}(y)))$.

From there one can work out the density by differentiation, leading to the standard result:

$$f_Y(y)= f_X(h^{-1}(y)) \left|\frac{d h^{-1}(y)}{dy}\right|$$

In other cases things are more complicated, but in some cases may still be doable.

• Perhaps you should add a fourth case right after the approximate normality of $X^2$ pointing out that the distribution is exactly normal if and only if $f(X) = aX+b$. – Dilip Sarwate May 9 '13 at 11:53
• Yes, indeed, it occurred to me that I should mention it shortly after I finished the previous edit ... but I couldn't come back to put it in until now. Thanks for the nudge. – Glen_b May 9 '13 at 17:51
• You are welcome. However, please note that @whuber has just pointed out in a comment on the main question that "if and only if" is incorrect; it should be "if" only. – Dilip Sarwate May 9 '13 at 17:56
• I hadn't seen those comments, but when I read your earlier comment on my answer it was immediately clear to me that a nonlinear transformation can still yield a normal; I have included an example in the edit to my answer that discusses the linear case (not the one @whuber mentioned though, I put in the simplest of several that occurred to me), though his example would have been a great one to include. I appreciate your comments. – Glen_b May 9 '13 at 18:10
• I changed the placement of the word "only" in the sentence about nonlinear transformations possibly resulting in normal random variables. I think it makes the sentence read better. Please put back the way it was if you don't think so too. – Dilip Sarwate May 9 '13 at 18:19