Probability help please?

Question

Suppose X is an exponential random variable with parameter p and c>0. Prove that cX is exponential with parameter p/c.

Answer 1

the CDF of the exponential distribution is:

P(X < x) = 1 - exp(-px)

P(cX < y) = P(cX < y)
= P(X < y/c) = 1 - exp(-p * y/c)
= 1 - exp(-p/c * y)

this shows that Y = cX has the exponential distribution with parameter p/c

Answer 2

We have
F_X(s) = P(X ≤ s) = 1 - e^(-ps), s ≥ 0 and
F_X(s) = 0, s < 0

so

F_(cX)(s) = P(cX ≤ s) = P(x ≤ (s/c)) = 1 - e^(-ps/c) = 1 - e^(-(p/c)s) for s ≥ 0
F_(cX)(s) = 0 for s < 0

which is the distribution function for an exponential random variable with parameter p/c