The question asked whether or not f(x) = 0.5e^x when x is less than or equal to zero and f(x) = 0.5e^(-x) when x is greater than or equal to zero is a valid probability density.

Question

I know that it is the integral... but usually you have a domain that is not a single point. Should I increase the domain slightly??

Answer 1

huh? Looks to me like the domain is all real numbers. The fact that x=0 is included in both parts doesn't really matter, since e^0 = e^-0 = 1

So, since the functions are symmetric, all you need is

∫[0,∞] 0.5 e^-x dx = 1/2

Answer 2

OH MY GOODNEss! I never even thought of that! Thank you so much!