The time to decay of an atom in a radioactive substance is a random variable X. The law of radioactive decay states that if N atoms are present at time t = 0, then N ƒ(t) atoms will be present at time t, where ƒ(t) = e^−kt (k > 0 is the decay constant). Explain the following statements:

(a) The fraction of atoms that decay in a small time interval [t, t + Δt] is approximately −ƒ'(t)Δt.

(b) The probability density function of X is y = −ƒ'(t).

(c) The average time to decay is 1/k.