"Let the random process K(t) depend on uniform Poisson process N(t), with the mean arrival rate λ>0, as follows: Starting at t = 0, both N(t) = 0, and K(t) = 0. When an arrival occurs in N(t), an independent Bernoulli trial takes place with probability of success p, where 0>p>1.
On success, K(t) is incremented by 1, otherwise K(t) is left unchanged. Find the PMF of the discrete valued random process K(t) at time t, that is, Pk (k ; t),for t ≥ 0."
So far, I've assumed the following...
The probability of k arrivals in interval t follows the distribution
P(k arrivals in time t)=((λt)k e-λt)/k!
And PMF of K(t) conditioned on a fixed number of k arrivals from the Poisson process
P(K(t) = i | K = k) = Ck,i(p)i(1-p)k-i
However, I am stuck trying to get the PMF of K(t) unconditioned on K.