Question

I . (20%) Random variable X has the probability density function as ; Random variable Y 2X+1 0 otherwise a) Determine A b) Determine the Probability Distribution Function F, (x) c) Determine E(X) and ơx d) Determine the probability density function fy(y) and E(Y)

Answer 1

Given that fx (x) = rightarrow A/4, 2 < x < 4 rightarrow 0 otherwise r = 2x + 1 (a) we know that - integral^infinity_-infinity fx (x) dx = 1 so integral^4_2 A/4 dx = 1 implies A/4 (x)^4 2 = 1 A/4 (4 - 2) = 1 implies A/4 times 2 = 1 A = 2 (b) we know that Fx (x) = = integral^x_-infinity fx (x) dx If x < 2 then fx (x) = 0 so Fx (x) = 0 If 2 < x < 4, fx (x) = 1/2 Fx (x) = integral^x_2 fx (x) dx Fx (x) = integral^x_2 1/2 dx = 1/2 (x)^x_2 Fx (x) = x - 2/2 If x < 4 then Fx (x) = integral^4_2 1/2 dx + integral^x_4 (0) dx Fx (x) = 1/2 [x]^4_2 + 0 so Fx (x) = 1/2 (4 - 2) = 1 so Fx (x) will be- (c) E [X] = integral^infinity_-infinity x fx (x) dx.....