Find the pdf corresponding to each of the following CDFs a F

Find the pdf corresponding to each of the following CDFs:

(a) F(x) = (x² + 2x + 1)/16; –1 <= x <= 3.

(b) F(x) = 1 – e^–(lambda)x – (lambda)xe^–(lambda)x; 0 <= x < infinity; lambda > 0.

Solution

a)

F(x) = (x² + 2x + 1)/16; –1 <= x <= 3.

=>PDF = f(x) = F\'(x) = (2x + 2)/16

=> f(x) = (x+1)/8 ; –1 <= x <= 3.

b)

F(x) = 1 - e^(-lambda*x) - lambda*x* e^(-lambda*x) ; 0 <= x <= infinity.

=>PDF = f(x) = F\'(x) = (lambda)e^(-lambda*x) - lambda(x *e^(-lambda*x)*-lambda + e^(-lambda*x) )

=>PDF = f(x) = F\'(x) = (lambda)e^(-lambda*x) - lambda(-lambda *x*e^(-lambda*x) + e^(-lambda*x) )

=>f(x) = (lambda)e^(-lambda*x) + lambda^2 *x*e^(-lambda*x) - (lambda)e^(-lambda*x)

=> f(x) = lambda^2 *x*e^(-lambda*x) ; 0 <= x <= infinity.