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Chapter 1: Q10E (page 1)

Question:Suppose that a point (X, Y ) is chosen at random from the circle S defined as follows:\({\rm{S}} = \left\{ {\left( {x,y} \right):{x^2} + {y^2} \le 1} \right\}\)

a. Determine the joint p.d.f. of X and Y , the marginal p.d.f. of X, and the marginal p.d.f. of Y .

b. Are X and Y independent

Short Answer

Expert verified

\(f\left( {x,y} \right)\)is constant over the circle S.

Area of S=\(\pi \) unit

Step by step solution

01

Given information

\(f\left( {x,y} \right)\)is constant over the circle S.

Area of S=\(\pi \) unit

02

Calculating pfd and marginal pdf.

a.

Area of s is\(\pi \)

By symmetry, the random variable Y will have the same marginal pdf as x:

03

Checking for independency of variables.

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