Question

Suppose the probability density of X is given by

f(x) = {(kxe)^(-x^2), X>0

=0, Otherwise

(a) find the value of k.

(b) find the distribution function of X, i.e., the cumulative density function of X.

Answer 1

a. $∫ _{ −∞}^ ∞ ​ f(x)dx=1$

So $\int_0^1kxe^{-x^2}dx=k(-e^{-x^2}/2)|_0^1=0.316k=1$

k=3.16

b.

$F(x)=\int_0^x3.16te^{-t^2}dt=3.16(-e^{-t^2}/2)|_0^x=3.16(-e^{-x^2}/2)-3.16/2+C$