0

I have 2D random walk and I would like to find out what distance I will travel after 200 steps. So I introduce two random variables $Z^{(200)}_x$ and $Z^{(200)}_y$ which tell me probabilities of my $x$ and $y$ position after 200 steps. Using CLT I can say that $$Z^{(200)}_x\approx \mathcal{N}(\mu,\sigma^2)\hspace{5pt}\text{and}\hspace{5pt}Z^{(200)}_y\approx \mathcal{N}(\mu,\sigma^2)$$ for some parameters $\mu$ and $\sigma$ (they are the same in $x$ and $y$). Since I am interested in the distance, I introduce new random variable $$Z^{(200)}=\sqrt{(Z^{(200)}_x)^2+(Z^{(200)}_y)^2}.$$ Using previous I can aproximate $Z^{(200)}$ and say that $$Z^{(200)}\approx\sqrt{(\mathcal{N}_1(\mu,\sigma^2))^2+(\mathcal{N}_2(\mu,\sigma^2))^2}.$$

Question

How to compute density function of random variable $\sqrt{(\mathcal{N}_1(\mu,\sigma^2))^2+(\mathcal{N}_2(\mu,\sigma^2))^2}?$ I am interested in finding out an explicit formula.

This may be stated more broadly, i.e. "how to compute density function of the square root of random variable?"

Fallen Apart
  • 3,785
  • See here: http://math.stackexchange.com/questions/1059938/whats-the-expectation-of-square-root-of-chi-square-variable/1060019#1060019 –  Mar 21 '16 at 20:50
  • You need Multi-Dimentional CLT. Let $Z_i$ be distribution of a single step. Then when properly scaled, $n^{-1/2}(S_n-E(Z))\to N(0,\Sigma)$. $P(\sqrt X\le t)=F_X(t^2)$. Differentiate to get density. – A.S. Mar 21 '16 at 20:51
  • @A.S. To make approximation of $Z^{(200)}$ precise or to compute $\sqrt{(\mathcal{N}_1(\mu,\sigma^2))^2+(\mathcal{N}_2(\mu,\sigma^2))^2}?$ – Fallen Apart Mar 21 '16 at 21:03
  • @ByronSchmuland "By the transformation technique, using the transformation $Y=g(Y)=\sqrt(Y)$, we have...". So there are general techniques to grasp such problems. I think I need to ivestigate them. – Fallen Apart Mar 21 '16 at 21:18

0 Answers0