Question

Let T be a continuous random variable with pdf

f(x) = (1/10(e)^-t/10 , : t ⩾ 0

(0, : x < 0

Define the continuous random variable by

(100, : 0 < T ⩽ 1

X= (50, : 1 < T ⩽ 3

(0, : T > 3

Find E(X)

Answer 1

we can use $∫^∞_{ −∞} fX(u)du=1:$

$1 =∫^∞_{−∞}fX(u)du\\ =∫^1_{−1}cu^2du\\ =\frac{2}{3}c.$

$EX =∫^1_{−1}uf_X(u)du\\ =\frac{3}{2}∫^1_{−1}u^3du\\ =0.$