Proving a random variable is an exponential distribution

Question

Suppose $X$ is a continuous random variable, with $$P(X>a+b)=P(X>a)P(X>b)\qquad \forall a,b>0.$$ Prove that $X$ is exponentially distributed.

I know this random variable must have the PDF $f_X(x)=\lambda e^{-\lambda x}$; how do I prove it given the above?

Answer 1

This is too long for a comment

1. From this you will see that any function that satisfies

$$ f(x + y) = f(x)f(y) $$

implies that

$$ f(x) = e^{\mu x} $$

2. Call $f(y) = P(X > y)$ in the expression above, we must then have something like

$$ P(X > y) = e^{\mu y} $$

But since we require that this number vanishes as we approach infinity, this implies that $\mu$ must be negative, $\mu = -\lambda$, for $\lambda > 0$

$$ P(X > y) = e^{-\lambda y} = \int_y^{\infty}{\rm d}y' f_X(y') $$

3. Applying the fundamental theorem of calculus you get

$$ f_X(x) = \lambda e^{-\lambda x} $$