Suppose X has density f(x) = c/(x^4) for x > 1, and f(x) = 0 otherwise, where c is a constant. Find a) c; b) E(X); c) Var(X).
I'm so confused...I dont even know where to start...
Thanks in advance!
So the integral of f(x) over all x equals 1 (since its a probability density function) so
$\displaystyle \int_{1}^{\infty} \frac{c}{x^4} dx=1$ so you need to integrate, then solve for c.
$\displaystyle E(X) = \int_{1}^{\infty} x \cdot \frac{c}{x^4} dx$
and
$\displaystyle Var(X) = E(X^2)-(E(X))^2$
So you can use a similiar process to find $\displaystyle E(X^2)=\int_{1}^{\infty} x^2 \cdot \frac{c}{x^4} dx$