Question

1. Let X and Y be independent random variables each uniformly distributed on (0,1) find
   1. a. P( |X-Y| < 0.5 )
   2. b. P( | X / Y -1 | < 0.5 )
   3. c. P( Y > X | Y > 1/2 )

Answer 1

Given that

let x,y be independent random variables each uniformly distributed on (0,1)

a) P(|X-Y|<0.5) = 0∫^0.5 0∫^0.5+Y (X-Y) dxdY +  0∫^0.5 0∫^0.5+x (y-x) dydx

⇒0∫^0.5+Y (X-Y) dx  =-0.5y²+0.125

⇒0∫^0.5(-0.5y²+0.125)dy = 0.04167

⇒0∫^0.5+x (y-x) dy = -0.5x²+0.125

⇒0∫^0.5( -0.5x²+0.125) dx = 0.04167

0∫^0.5 0∫^0.5+Y (X-Y) dxdY +  0∫^0.5 0∫^0.5+x (y-x) dydx = 0.04167+0.04167 = 0.08334

P(|X-Y|<0.5) = 0.08334

b) P (|X/Y-1|<0.5) =  ₀∫^0.5₀∫^0.5X(X-Y) dydx +  ₀∫^0.5₀∫^0.5Y (y-x) dXdY

₀∫^0.5X(X-Y) dY = 0.375x²

₀∫^0.5(0.375x²) dy = 0.015625

₀∫^0.5Y (y-x) dX = 0.375y²

₀∫^0.5 (0.375y²) dy = 0.015625

₀∫^0.5₀∫^0.5X(X-Y) dydx +  ₀∫^0.5₀∫^0.5Y (y-x) dXdY = 0.015625 + 0.015625 = 0.03125

P (|X/Y-1|<0.5) = 0.03125