Can Somebody HEEELP PLEASE!!! ASAP

Question

Definition: A function p (t) is a probability density function provided it satisfies the following properties:

1. p (t) >= 0 for all t,

2. integral from (negative infinity ) to ( infinity) of p(t) dt = 1

Cells in a tissue culture are treated with an experimental medicine. Let t be the survival time of the cells in hours after receiving the treatment. The probability density function which describes this situation is

p (t) ={ Ce^-2t , if 0<=t<= infinity,

{ 0 , if t < 0.

Find the value of the constant C with detailed steps.

Answer 1

Since the given function is zero for negative t, we only need to integrate from zero to infinity to find C.

∫₀^∞Ce^-2tdt = 1

(Ce^-2t)/(-2) |₀^∞ = 1

(-C/2)(0 - 1) = 1

C/2 = 1

C = 2