How to Find EX and Var(X) for a Continuous Random Variable with Given P(X>x)?

[badgerbadger]

Let X be a continuous random variable with

P(X>x)=e^-ax, x less/equal 0

Where a is a positive constant. Find EX and Var(x)

 

[HallsofIvy]

If P(X> x)= e^{-ax}, x less than or equal to 0, then the probability density function is p(x)= -a e^{-ax}.

Now just use the definitions:
[tex]E(x)= -a \int_x^0 xe^{-ax}dx[/itex]
and
$$Var(x)= -a \int_x^0 (x- E(x))^2e^{-ax}dx= -aE(x)\int_x^0 x^2e^{-ax}- E^2(x)$$