This is a math question about probability density function which is continuous. I would really appreciate help for this question.
Let $X$ be a continuous random variable which follows the following probability
density function:
$$f(x) = e^{-x} \quad \quad when \quad x\ge0 \\\\f(x)=0 \qquad \quad \quad otherwise$$
1) For $n=${$1,2,3,…$}, establish a relationship between $E(X^{n})$ and $E(X^{n-1})$. Hence find $E(X^n)$.
2) Find $E(X(X + 1)(X – 1))$ and $Var(4X + 1)$.
I have difficulty with the integration for $E(X^n)$. I'm not sure what to the with the power of n after doing integration by parts. I'm pretty sure it's linked to the $E(X^{n-1})$, but I don't know how so.
And for the second question, I don't even know from where to start.
Many thanks in advance!
Best Answer
Hint: Note that $E(X^n) = \int_0^{\infty} x^n e^{-x} dx = n\int_0^{\infty} x^{n-1} e^{-x} dx =n E(X^{n-1}) $