Question

Find the expectation of a random variable X if f(x) = ce⁻ ˣ for x>0 and 0 otherwise.

Answer 1

Solution:

$\int_0^{\infty} f(x)dx=1 \\\Rightarrow\int_0^{\infty} (ce^{-x})dx=1 \\\Rightarrow -c[e^{-x}]_0^{\infty}=1 \\\Rightarrow -c[0-1]=1 \\\Rightarrow c=1$

Then,

$E[X]=\int_0^{\infty}x f(x)dx \\=\int_0^{\infty}x (ce^{-x})dx \\=\int_0^{\infty}x e^{-x}dx \\=[-e^{-x}x-e^{-x}]_0^{\infty} \\=[0-0]-[0-1] \\=1$