Chapter 6: Q.6.52 (page 274)

Let X and Y denote the coordinates of a point uniformly chosen in the circle of radius $1$ centered at the origin. That is, their joint density is $f (x, y) = \frac{1}{π} x^{2} + y^{2} \leq 1$ .
Find the joint density function of the polar coordinates $R = {(X^{2} + Y^{2})}^{1 / 2}$ and $θ = \tan^{- 1} Y / X$ .

Short Answer

Expert verified

Joint density function of the polar coordinates,

$f_{R, θ} (r, θ) = \frac{r}{π}$

Step by step solution

Radius :

A straight line drawn from the center of a circle or sphere to the circumference.

Explanation :

The random vector's density function,

$f (x, y) = \frac{1}{π}$

For $x^{2} + y^{2} \leq 1$

Polar coordinates: $R = {(X^{2} + Y^{2})}^{1 / 2}$ and $θ = \tan^{- 1} Y / X$ .

The density function of a random vector, according to the assertion,

$f_{X . Y} (x, y) = \frac{1}{π}$ for $x^{2} + y^{2} \leq 1$ .

Now, apply the transformation

$g (x, y) = (r, θ) = (\sqrt{x^{2} + y^{2}}, \tan^{- 1} \frac{y}{x})$

We can express the density function of a random vector using the theorem about variable transformations $(R, θ) = g (X, Y)$ as

$f_{R, θ} (r, θ) = f_{X, Y} (x, y) . {|d e t (\nabla g (x, y))|}^{- 1}$

Then calculate,

$\nabla g (x, y) = [\begin{matrix} \frac{x}{\sqrt{x^{2} + y^{2}}} & \frac{y}{\sqrt{x^{2} + y^{2}}} \\ - \frac{y}{x^{2} + y^{2}} & \frac{x}{x^{2} + y^{2}} \end{matrix}]$

That yields

$|d e t (\nabla g (x, y))| = \frac{x^{2} + y^{2}}{{(x^{2} + y^{2})}^{\frac{3}{2}}} = \frac{1}{\sqrt{x^{2} + y^{2}}} = \frac{1}{r}$

Hence,

$f_{R, θ} (r, θ) = r . f_{X . Y} (x, y) = \frac{r}{π}$ for $r \in (0, 1)$

Androle="math" localid="1647253987981" $θ \in (0, 2 π)$

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Short Answer

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Let X and Y denote the coordinates of a point uniformly chosen in the circle of radius 1centered at the origin. That is, their joint density is f(x,y)=1πx2+y2≤1.Find the joint density function of the polar coordinates R=(X2+Y2)1/2 and θ=tan-1Y/X.

Short Answer

Step by step solution

Radius :

Explanation :

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

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