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The relationship between the standard normal and the chi-squared distributions is well known. I was wondering though, is there a transformation that can lead from a $\chi^2 (1)$ back to a standard normal distribution?

It can be easily seen that the square root transformation does not work as its range is only positive numbers. I believe the resulting distribution is called folded normal. Is there a clever trick that works here?

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One option is to exploit the fact that for any continuous random variable $X$ then $F_X(X)$ is uniform (rectangular) on [0, 1]. Then a second transformation using an inverse CDF can produce a continuous random variable with the desired distribution - nothing special about chi squared to normal here. @Glen_b has more detail in his answer.

If you want to do something weird and wonderful, in between those two transformations you could apply a third transformation that maps uniform variables on [0, 1] to other uniform variables on [0, 1]. For example, $u \mapsto 1 - u$, or $u \mapsto u + k \mod 1$ for any $k \in \mathbb{R}$, or even $u \mapsto u + 0.5$ for $u \in [0, 0.5]$ and $u \mapsto 1 - u$ for $u \in (0.5, 1]$.

But if we want a monotone transformation from $X \sim \chi^2_1$ to $Y \sim \mathcal{N}(0,1)$ then we need their corresponding quantiles to be mapped to each other. The following graphs with shaded deciles illustrate the point; note that I have had to cut off the display of the $\chi^2_1$ density near zero.

Chi squared distribution with one degree of freedom and deciles shaded Standard normal distribution with deciles shaded

For the monotonically increasing transformation, that maps dark red to dark red and so on, you would use $Y = \Phi^{-1}(F_{\chi^2_1}(X))$. For the monotonically decreasing transformation, that maps dark red to dark blue and so on, you could use the mapping $u \mapsto 1-u$ before applying the inverse CDF, so $Y = \Phi^{-1}(1 - F_{\chi^2_1}(X))$. Here's what the relationship between $X$ and $Y$ for the increasing transformation looks like, which also gives a clue how bunched up the quantiles for the chi-squared distribution were on the far left!

Mapping from chi squared with 1 df to standard normal

If you want to salvage the square root transform on $X \sim \chi^2_1$, one option is to use a Rademacher random variable $W$. The Rademacher distribution is discrete, with $$\mathsf{P}(W = -1) = \mathsf{P}(W = 1) = \frac{1}{2}$$

It is essentially a Bernoulli with $p = \frac{1}{2}$ that has been transformed by stretching by a scale factor of two then subtracting one. Now $W\sqrt{X}$ is standard normal — effectively we are deciding at random whether to take the positive or negative root!

It's cheating a little since it is really a transformation of $(W, X)$ not $X$ alone. But I thought it worth mentioning since it seems in the spirit of the question, and a stream of Rademacher variables is easy enough to generate. Incidentally, $Z$ and $WZ$ would be another example of uncorrelated but dependent normal variables. Here's a graph showing where the deciles of the original $\chi^2_1$ get mapped to; remember that anything on the right side of zero is where $W = 1$ and the left side is $W = -1$. Note how values around zero are mapped from low values of $X$ and the tails (both left and right extremes) are mapped from the large values of $X$.

Mapping chi-squared to normal distribution

Code for plots (see also this Stack Overflow post):

require(ggplot2)
delta     <- 0.0001 #smaller for smoother curves but longer plot times
quantiles <- 10    #10 for deciles, 4 for quartiles, do play and have fun!

chisq.df <- data.frame(x = seq(from=0.01, to=5, by=delta)) #avoid near 0 due to spike in pdf
chisq.df$pdf <- dchisq(chisq.df$x, df=1)
chisq.df$qt <- cut(pchisq(chisq.df$x, df=1), breaks=quantiles, labels=F)
ggplot(chisq.df, aes(x=x, y=pdf)) +
  geom_area(aes(group=qt, fill=qt), color="black", size = 0.5) +
  scale_fill_gradient2(midpoint=median(unique(chisq.df$qt)), guide="none") +
  theme_bw() + xlab("x")

z.df     <- data.frame(x = seq(from=-3, to=3, by=delta))
z.df$pdf <- dnorm(z.df$x)
z.df$qt  <- cut(pnorm(z.df$x),breaks=quantiles,labels=F)
ggplot(z.df, aes(x=x,y=pdf)) +
  geom_area(aes(group=qt, fill=qt), color="black", size = 0.5) +
  scale_fill_gradient2(midpoint=median(unique(z.df$qt)), guide="none") +
  theme_bw() + xlab("y")

#y as function of x
data.df <- data.frame(x=c(seq(from=0, to=6, by=delta)))
data.df$y <- qnorm(pchisq(data.df$x, df=1))
ggplot(data.df, aes(x,y)) + theme_bw() + geom_line()

#because a chi-squared quartile maps to both left and right areas, take care with plotting order
z.df$qt2 <- cut(pchisq(z.df$x^2, df=1), breaks=quantiles, labels=F) 
z.df$w <- as.factor(ifelse(z.df$x >= 0, 1, -1))
ggplot(z.df, aes(x=x,y=pdf)) +
  geom_area(data=z.df[z.df$x > 0 | z.df$qt2 == 1,], aes(group=qt2, fill=qt2), color="black", size = 0.5) +
  geom_area(data=z.df[z.df$x <0 & z.df$qt2 > 1,], aes(group=qt2, fill=qt2), color="black", size = 0.5) +
  scale_fill_gradient2(midpoint=median(unique(z.df$qt)), guide="none") +
  theme_bw() + xlab("y")
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    $\begingroup$ This is a very clever trick, I like it :). Thank you. $\endgroup$ – JohnK Nov 27 '14 at 16:55
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[Well, I couldn't locate the duplicate that I thought there was; the nearest I came was the mention of the fact toward the end of this answer. (It's possible it was only discussed in comments on some question, but perhaps there was a duplicate and I just missed it.) I'll give an answer here after all.]

If $X$ is chi-square, with $F$ as its CDF, and $Φ$ is the cdf of the normal, then $Φ^{−1}(F(X))$ is normal. This is obvious since the probability integral transform of $X$ gives a uniform, and $Φ^{−1}(U)$ is normal. So we have a monotonic transformation of the chi-squared to normal.

The same trick works with any two continuous variables.

This gives us a neat counterexample to the various versions of the question "are uncorrelated normal Y,Z bivariate normal?" that come up, since if Z is standard normal and $Y=Φ^{−1}(F_{χ^2_1}(Z^2))$, then $Z,Y$ are both normal and they're uncorrelated, but they're definitely dependent (and have a rather pretty bivariate relationship)

The transformation $T(z)=Φ^{−1}(F_{χ^2_1}(z^2))$:

enter image description here

Histogram of a large sample of $Z+Y$ values:

enter image description here

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  • $\begingroup$ Could you add just a few words to your answer to explain why $Z$ and $Y$ are uncorrelated? It is not at all obvious to me, but then, probably I am just not seeing things from the correct perspective. $\endgroup$ – Dilip Sarwate Nov 27 '14 at 3:15
  • $\begingroup$ @Dilip $Z$ is centered at 0 and the transformation of $Z$ we apply to get $Y$ is symmetric about 0. So $E(YZ)=0$. $\endgroup$ – Glen_b Nov 27 '14 at 4:23
  • $\begingroup$ Well, we need to show a little more to be sure E(YZ) is finite, for example, but that would present no problem. $\endgroup$ – Glen_b Nov 27 '14 at 7:59
  • $\begingroup$ The mapping $z \mapsto T(z)$ can also be seen nicely in the normal density plots in my answer - the bottom one (red in middle) maps to the top one (red on left), with red mapped to red and blue mapped to blue. It is a pretty relationship! $\endgroup$ – Silverfish Nov 29 '14 at 0:38

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