Suppose that XP BernoulliP and P Uniform01 How do I find t

Suppose that X|P ~ Bernoulli(P) and P ~ Uniform(0,1). How do I find the marginal distribution of X?

Solution

Assuming that
X= 1 when the outcome is a success
X = 0 when outcome is a failure
If p is the probability of a success then the pmf is, p(0) =P(X=0) =1-p p(1) =P(X=1) =p

P is a Bernoulli random variable if it has the above pmf for p between 0 and 1.

Expected value of Bernoulli r. v.:
E(X) = 0*(1-p) + 1*p = p

Variance of Bernoulli r. v.:
E(X²) = 0²*(1-p) + 1²*p = p
Var(X) = E(X²) - (E(X))² = p - p² = p(1-p)

------------------

For multiple trials

For each i{1,2,…,k}, Yi has the binomial distribution with parameters n and p_i:

P(Yi=j) = (n_j)p_ji(1p_i)^nj, j{0,1,…,n}

There is a simple probabilistic proof. If we think of each trial as resulting in outcome i or not, then clearly we have a sequence of n Bernoulli trials with success parameter pi.

Random variable Y_i is the number of successes in the n trials.