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# ISI MStat PSB 2009 Problem 4 | Polarized to Normal

This is a very beautiful sample problem from ISI MStat PSB 2009 Problem 4. It is based on the idea of Polar Transformations, but need a good deal of observation o realize that. Give it a Try it !

## Problem- ISI MStat PSB 2009 Problem 4

Let $$R$$ and $$\theta$$ be independent and non-negative random variables such that $$R^2 \sim {\chi_2}^2$$ and $$\theta \sim U(0,2\pi)$$. Fix $$\theta_o \in (0,2\pi)$$. Find the distribution of $$R\sin(\theta+\theta_o)$$.

### Prerequisites

Convolution

Polar Transformation

Normal Distribution

## Solution :

This problem may get nasty, if one try to find the required distribution, by the so-called CDF method. Its better to observe a bit, before moving forward!! Recall how we derive the probability distribution of the sample variance of a sample from a normal population ??

Yes, you are thinking right, we need to use Polar Transformation !!

But, before transforming lets make some modifications, to reduce future complications,

Given, $$\theta \sim U(0,2\pi)$$ and $$\theta_o$$ is some fixed number in $$(0,2\pi)$$, so, let $$Z=\theta+\theta_o \sim U(\theta_o,2\pi +\theta_o)$$.

Hence, we need to find the distribution of $$R\sin Z$$. Now, from the given and modified information the joint pdf of $$R^2$$ and $$Z$$ are,

$$f_{R^2,Z}(r,z)=\frac{r}{2\pi}exp(-\frac{r^2}{2}) \ \ R>0, \theta_o \le z \le 2\pi +\theta_o$$

Now, let the transformation be $$(R,Z) \to (X,Y)$$,

$$X=R\cos Z \\ Y=R\sin Z$$, Also, here $$X,Y \in \mathbb{R}$$

Hence, $$R^2=X^2+Y^2 \\ Z= \tan^{-1} (\frac{Y}{X})$$

Hence, verify the Jacobian of the transformation $$J(\frac{r,z}{x,y})=\frac{1}{r}$$.

Hence, the joint pdf of $$X$$ and $$Y$$ is,

$$f_{X,Y}(xy)=f_{R,Z}(x^2+y^2, \tan^{-1}(\frac{y}{x})) J(\frac{r,z}{x,y}) \\ =\frac{1}{2\pi}exp(-\frac{x^2+y^2}{2})$$ , $$x,y \in \mathbb{R}$$.

Yeah, Now it is looking familiar right !!

Since, we need the distribution of $$Y=R\sin Z=R\sin(\theta+\theta_o)$$, we integrate $$f_{X,Y}$$ w.r.t to $$X$$ over the real line, and we will end up with, the conclusion that,

$$R\sin(\theta+\theta_o) \sim N(0,1)$$. Hence, We are done !!

## Food For Thought

From the above solution, the distribution of $$R\cos(\theta+\theta_o)$$ is also determinable right !! Can you go further investigating the occurrence pattern of $$\tan(\theta+\theta_o)$$ ?? $$R$$ and $$\theta$$ are the same variables as defined in the question.

Give it a try !!

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