probability density function

Question

Let X be the amount of time (years) before a new battery stops working. X has the probability density function

f(t) = { C e^(-0.4t) , t ≥ 0

0 , otherwise,

where C is a constant.

Find C (using (Pr(-inf < x < inf) = 1))

Answer 1

Hi, Juyeon. To solve this problem, you have to recall the property of PDFs that the integral from -∞ to ∞ of the PDF is equal to 1. Note that the integral over the region where the PDF is 0 is just 0, so you can remove that region from the bounds of your integral. Thus, you have the equation ∫₀^∞ C e^-.4t dt = 1. Evaluating the integral leaves you with an equation such that C is the only unknown. At this point, solve for C.