MATH CALCULUS2 HARD QUESTION

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 .

Answer 1

You will need to determine C by making use of ∫_(0,∞) C*e^-2t dt = 1:

∫_(0,∞) C*e^-2t dt = (-C/2)*e^-2t | ₀ ^∞ = 0 - (-C/2) = C/2 = 1 ==> C = 2