Question

If X ~U(-2; 4), find the probability density function of the random variable:

a) Y = 2X + 3.

b) Y = 1/(X+2)⁴

c) Y = X²u(X), where u(x) is the unit step function. Hint: first sketch g(x) = x²u(x)

Answer 1

Given problem has been solved with proper explanation and calculations. To understand this problem, a basic knowledge of uniform distribution is required. If you have any doubt, please comment.

If Xnu (-2;4), Find the probability dencity function of the random variable: - First of all, xau (-2, 4) denotes a uniform distribution. f(x) Xnu(a,b) denotes Ta b • lower limit of x is a f(x) = probability dencity • upper limit ofis b function of a f(x) = H bora » asasb i asxsb ( 0 ; otherwise (a) Y=2x+8 where, &nu (-2,4) & when x = -2, then y = +2(-2) + 3 = -1 When x = 4, then Y = 2(4) + 3 = 11 in you(-1,11) & lower limit of Y is – 1 upper limit of Yes 11 b = +11 a: -1 + Probability denuty function of Yes f(y)-S 6-ate; -}<43! o ; otherwise.

Uniform distribution of the given random variable. (b) y = x + 2) where x~0 (-2,4) When X = -2, then y = undefined. < y *** This randos variable is undefined. (c) y = x 2 u(x) where u(x) is unit step function. - unet step function definètion: u(x) = 1 ; *>,0 = 0 ; x<0 When x = -2, then y=-2 [2u (-2)] = 0 [ As u(+2) =0) . When x = 4, then y= 4 [2u(4)] = 4x2 = 8 [As U(4)-1] e You (0,8) where aco, b28 *** Probability density function of yes, so f(y) = S bra § 3 03438 l o & otherwise to tk