Question

Let X and Y be geometric random variables with parameters 0.2 and 0.4. Find the Distribution of min(X,Y). Please show all work.

Answer 1

X ~ Geom(p = 0.2) and Y ~ Geom(p = 0.4)

CDF of X and Y are given as,

P(X > x) = (1 - 0.2)^x = 0.8^x

P(Y > y) = (1 - 0.4)^y = 0.6^y

Let Z = min(X,Y).

CDF of Z is,

P(Z z) = 1 - P(Z > z) = 1 - P(min(X, Y) > z)

= 1 - P(X > z, Y > z)

= 1 - P(X > z) P(Y > z) (X and Y are independent random variables )

= 1 - 0.8^z * 0.6^z

= 1 - 0.48^z

Thus,

P(Z z) = 1 - 0.48^z

P(Z > z) = 0.48^z   = (1 - 0.52)^z

which is the CDF of  geometric random variables with parameters 0.52

Thus,   min(X,Y) ~ Geom(p = 0.52)