This is an interesting problem from intertwined conditional probability and Bernoulli random variable mixture, which gives a sweet and sour taste to Problem 4 of ISI MStat 2016 PSB.
Let and
be three Bernoulli
random variables such that
and
are independent,
and
are independent, and
and
are independent.
(a) Show that .
(b) Show that if equality holds in (a), then
We use the fact that and
are independent,
and
are independent, and
and
are independent.
and
be three Bernoulli
random variables. Hence,
.
.
Now, this is just a logical game with conditional probability.
.
.
.
.
Now, is a Bernoulli
random variable. So,
.
.
.
.
.
Hence,
This is an interesting problem from intertwined conditional probability and Bernoulli random variable mixture, which gives a sweet and sour taste to Problem 4 of ISI MStat 2016 PSB.
Let and
be three Bernoulli
random variables such that
and
are independent,
and
are independent, and
and
are independent.
(a) Show that .
(b) Show that if equality holds in (a), then
We use the fact that and
are independent,
and
are independent, and
and
are independent.
and
be three Bernoulli
random variables. Hence,
.
.
Now, this is just a logical game with conditional probability.
.
.
.
.
Now, is a Bernoulli
random variable. So,
.
.
.
.
.
Hence,