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I have a random variable $X$ which follows the following probability density function,

$$ p(x) = \frac{1}{4\pi} \Big[ \operatorname{erf}\Big(\frac{k\mu-x+2\pi}{\sqrt{2}k\sigma}\Big) - \operatorname{erf}\Big(\frac{k\mu-x}{\sqrt{2}k\sigma}\Big) \Big]$$

I wanted to find the probability density function for the function of the random variable $Y = \cos(X)$.

I followed the CDF method to find this.

$$ F_Y(y) = p(Y \leq y) = p(\cos(X) \leq y) = p(X \leq \cos^{-1}(y))$$

$$ F(y) = \int_{-\infty}^{\cos^{-1}(y)} \frac{1}{4\pi} \Big[ \operatorname{erf}\Big(\frac{k\mu-x+2\pi}{\sqrt{2}k\sigma}\Big) - \operatorname{erf}\Big(\frac{k\mu-x}{\sqrt{2}k\sigma}\Big) \Big] dx $$

The terms $\mu$, $\sigma$ and $k$ are constants.

$$ k \quad \epsilon \quad Z; k \geq 0$$ $$ \sigma \quad \epsilon \quad R; \sigma > 0 $$

Mathematica Output: $$ F_Y(y) = \frac{1}{8 \pi ^{3/2}} \Bigg[ -\sqrt{\pi } \left(2 i \log \left(\sqrt{1-y^2}+i y\right)-2 k \mu+\pi \right) \text{erf}\left(\frac{-2 i \log \left(\sqrt{1-y^2}+i y\right)+2 k \mu-\pi }{2 \sqrt{2} k \sigma}\right) \\ +\sqrt{\pi } \left(2 i \log \left(\sqrt{1-y^2}+i y\right) -2 k \mu-3 \pi \right) \text{erf}\left(\frac{-2 i \log \left(\sqrt{1-y^2}+i y\right) +2 k \mu+3 \pi }{2 \sqrt{2} k \sigma}\right)- \\ 2 \sqrt{2} k \sigma \left(\sqrt{1-y^2}+i y\right)^{\frac{i (2 k \mu+3 \pi )}{2 k^2 \sigma^2}} \exp \left(-\frac{(2 k \mu+3 \pi )^2-4 \log ^2\left(\sqrt{1-y^2}+i y\right)}{8 k^2 \sigma^2}\right)+ \\ 2 \sqrt{2} k \sigma \left(\sqrt{1-y^2}+i y\right)^{-\frac{i (\pi -2 k \mu)}{2 k^2 \sigma^2}} \exp \left(-\frac{(\pi -2 k \mu)^2-4 \log ^2\left(\sqrt{1-y^2}+i y\right)}{8 k^2 \sigma^2}\right)+4 \pi ^{3/2} \Bigg] $$

When I do this integral, it gives me a complex valued function on mathematica. How can this be possible that the CDF is complex valued? If it is so, what should I consider as the CDF? Should I consider the real part only?

============== EDIT ==========

How I found $p(x)$ ?

It comes from another problem of finding the PDF of a random variable $ X = k U + \beta $, where k is an integer. $U$ comes from a Gaussian distribution and $\beta$ comes from a uniform distribution.

$$ U \sim \mathcal{N}(\mu, \sigma) $$ and $$ \beta \sim \mathcal{U}(0, 2\pi) $$

As it is an addition, I found the PDF of $X$ as the convolution of the other 2 PDFs.

$$ p(x) = \int_{0}^{2\pi} \frac{1}{2\pi \sqrt{2 \pi k^2 \sigma^2}} e^{-(kx - k\mu - \tau)/(2k^2\sigma^2)} d\tau $$

$$ p(x) = \frac{1}{4\pi} \Big[ \operatorname{erf}\Big(\frac{k\mu-x+2\pi}{\sqrt{2}k\sigma}\Big) - \operatorname{erf}\Big(\frac{k\mu-x}{\sqrt{2}k\sigma}\Big) \Big] $$

I thought this was correct, because when I integrated $p(x)$ from $-\infty$ to $+\infty$, I got $1$.

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    $\begingroup$ $p(\cos(X) \leq y) = p(X \leq \cos^{-1}(y))$ looks a dubious statement if $X$ can take any real value. Even if $X$ is restricted to $[0,\pi]$ and if $y \in [-1,1]$, then $\cos^{-1}(y)$ is a decreasing function $\endgroup$
    – Henry
    Commented Aug 11, 2022 at 12:25
  • $\begingroup$ I just applied cosine inverse on both sides to see how can I compute the CDF with $p(x)$. If this step is wrong, how to proceed? $X$ an take any value but a cosine function limits it from -1 to 1. $\endgroup$
    – CfourPiO
    Commented Aug 11, 2022 at 12:29
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    $\begingroup$ Answer to the question in title: No, it is not possible. Mathemathica treats erf as a complex function, can you tell mathematica not to do that? $\endgroup$ Commented Aug 11, 2022 at 13:08
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    $\begingroup$ @Kjetil Yes, you can. It's important to simplify complex-looking output when you know it's real. Mathematica has gradually introduced some bugs in integration (especially after version 7 or so) and one has to take care to test the output. $\endgroup$
    – whuber
    Commented Aug 11, 2022 at 15:37

1 Answer 1

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It is not obvious to me that your $p(x)$ is actually a density, but let's suppose it is. Meanwhile $\cos^{-1}$ is not defined on all the reals and is usually a decreasing function $[-1,1] \to [0,\pi]$.

For $y \le -1$ you have $F_Y(y) = \mathbb P(Y \leq y) = 0$

and similarly for $y \ge 1$ you have $F_Y(y) = \mathbb P(Y \leq y) = 1$.

Otherwise $-1<y<1$, and with $\cos^{-1}: [-1,1] \to [0,\pi]$ you have

$$F_Y(y) = \mathbb P(Y \leq y) = \mathbb P(\cos(X) \leq y) \\= \mathbb P(2n\pi +\cos^{-1}(y)\le X \le 2n\pi+2\pi-\cos^{-1}(y))$$ over integer $n$ and so

$$F_Y(y) = \sum_{n=-\infty}^{\infty} \int_{x=2n\pi +\cos^{-1}(y)}^{x=2n\pi+2\pi-\cos^{-1}(y)} p(x)\, dx$$

all of which should be real, though not necessarily simple

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  • $\begingroup$ I see! I understand it now clearly. Thank you. I was wondering why you said $p(x)$ doesn't look like a density. Is there a reason for it? I want to know because I derived it from another problem. I will update in the question now. $\endgroup$
    – CfourPiO
    Commented Aug 11, 2022 at 14:41
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    $\begingroup$ I can see $p(x)$ is positive, but it seems to involve the error function which is already an integral. I would not want to try to integrate it again to show $\int_{-\infty}^{\infty} p(x)\, dx =1$. But it might be OK $\endgroup$
    – Henry
    Commented Aug 11, 2022 at 15:01
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    $\begingroup$ Re "not obviously a density:" When $F$ is any distribution function and $a$ is any positive constant, $x\to F(x+a)-F(x)$ is a distribution function, too. It is the distribution of $X+U$ where $X$ is a random variable with distribution $F$ and $U$ is a Uniform$(0,a)$ variable. See the section on "Intuition from Probability" at stats.stackexchange.com/a/43075/919. $\endgroup$
    – whuber
    Commented Aug 12, 2022 at 12:52
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    $\begingroup$ @whuber Did you intend to say $\frac1a (F(x)-F(x-a))$ is a density? I am prepared to accept this, and also that the original expression is an example of this - I was just saying that it was not obvious $\endgroup$
    – Henry
    Commented Aug 12, 2022 at 13:10
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    $\begingroup$ Yes, I neglected the $1/a$ factor. (I always think in terms of convenient units of measurement, which in this case makes $a=1,$ and forgot to adjust for that. Thanks for catching the error!) $\endgroup$
    – whuber
    Commented Aug 12, 2022 at 13:15

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