A Bernoulli random variable is one that assumes only two values, 0 and 1 with

p(1) = p

and

p(0) = 1 p = q.

Show that this function has the properties of a distribution function.First note that

F(y) =

for

y < 0.

Thus,

limyF(y) = .

Next observe that

F(y) =

for

y > 1

and so

limyF(y) = .

Let

F(0) = q

and

F(1) = 1.

Since

p + q = 1

and p 0, we know

F(0) = q ? = 1 = F(1)

and therefore

F(0) ? = F(1).

Finally, combining our observations we can conclude that

F(y₁) ? = F(y₂),

for any

y₁ < y₂.