Question

A point (a, b) is distributed uniformly in the square 0<x<1, 0<y<1. Let S(a, b) be the area of a rectangle with sides a and b. Find P{1/4 < S(a, b) < 1/3}

Answer 1

point (a,b) is distributed uniformly in the square 0<x<1, 0<y<1.

S(a,b) be the area of a rectangle with sides a and b. we know that the area of rectangle is given by (length*breadth).

so area of rectangle is S(a,b)=ab

f(a,b) = 1    (since pdf of uniform distribution U[0,1])

now our aim is to find pdf of S(a,b)= ab

here we have 2 variables a and b which varies from 0 to 1.

so using joint probability distribution function

let    ab=t,    b=u

implies that a=t/u, b=u

which implies that u varies from t to 1.     

find jacobian J

pdf of joint variables t and u is given by

marginal distribution of t

now we calculate the probability

so