Question

If X is Rayleigh distributed amplitude with an average power of 10, what is the pdf of Y-1-exp(-XA2/10)? If X is N(0,4) and Y is N(0,9) and X and Y are independent, what is the prob (XY>0)?

Answer 1

Average power, is same as variance n sigma^2 = 10 PDF of X is f_X (x) = x/10 e^-x^2/20 y = 1 - e^-x^2/10 , 0 < y < 1 CDF of Y is F_Y (y) = Pr {Y lessthanorequalto y} = Pr{1 - e^-X^2/10 lessthanorequalto y} = Pr { -e^-X^2/10 lessthanorwequalto y - 1} = Pr { e^-X^2/10 greaterthanorequalto -y + 1} = Pr {-X^2/10 greaterthanorequalto ln(1 - y) } = Pr {X^2 lessthanorequalto -10 ln(1 - y) } = Pr (X lessthanorequalto squareroot -10 ln(1 - y) } , 0 < y < 1 squareroot -10 ln(1 - y) = integral_0 x/10 e^-x^2/20 dx, (Because X greaterthanorequalto 0) Let x^2/20 = t x dx/10 = dt when x = squareroot -10 ln (1 - y) t = (squareroot -10 ln (1 - y)^2/20 F_Y(y) = integral_0^-1/2 ln (1- y) e^-t dt = 1 - e^1/2 ln (1 - y) = 1 - e^ln (1 - y)^1/2 = 1 - squareroot 1 - y in PDF of Y is F_Y(y) = d/dy F_Y(Y) = 1 /2 squareroot 1 - y, 0 < y < 1