# Question: Suppose a Bernoulli trial results in a success with probability

Suppose a Bernoulli trial results in a success with probability p and a failure with probability 1 – p. Suppose the Bernoulli trial is repeated indefinitely with each repitition independent of all others. Let Xn be a “ success runs” Markov chain where represents the number of most recent consecutive successes that have been observed at the n th trial. That is Xn = m, if trial numbers n, n– 1, n – 2,…, n– m+ 1 were all successes but trial number n–m was a failure. Note that Xn = 0 if the th trial was a failure.

(a) Find an expression for the one- step transition probabilities, p i, j.

(b) Find an expression for the - step first return probabilities for state 0,f0(n)0 .

(c) Prove that state 0 is recurrent for any 0 < p < 1. Note that since all states communicate with one another, this result together with the result of Exercise 9.29 is sufficient to show that all states are recurrent.

(a) Find an expression for the one- step transition probabilities, p i, j.

(b) Find an expression for the - step first return probabilities for state 0,f0(n)0 .

(c) Prove that state 0 is recurrent for any 0 < p < 1. Note that since all states communicate with one another, this result together with the result of Exercise 9.29 is sufficient to show that all states are recurrent.

**View Solution:**## Answer to relevant Questions

Find the steady- state distribution of the success runs Markov chain. Suppose a Bernoulli trial results in a success with probability p and a failure with probability 1 – p. Suppose the Bernoulli trial is repeated ...An Inventory Model - A hot dog vendor operates a stand where the number of hot dogs he sells each day is modeled as a Poisson random variable with a mean value of 100. Let X [k] represents the number of hot dogs the vendor ...A sinusoidal signal of the form X (t) = bcos (2πft + θ), is transmitted from a fixed platform. The signal is received by an antenna which is on a mobile platform that is in motion relative to the transmitter, with a ...Find the PSD of the process Let Wn be an IID sequence of zero- mean Gaussian random variables with variance. Define a discrete- time random process, X[ n] = pX[ n – 1]+ Wn, n = 1, 2, 3, … where X[ 0] = W0 and is a ...Determine whether or not the periodogram is an unbiased estimate of the PSD.Post your question