Consider a renewal process for which the lifetimes X1, X2, . . . are iid Poisson random variables with rate \ be the time of the fith renewal. (a) Find the probability mass function of W5 as a function of n: P(W5 = n) = (b) Compute E[W5] = = 0.5. Let W5

Let X ~ Geom(p), a geometric distribution with parameter p € (0, 1). This is a discrete probability distribution on the positive integers, with p.m.f. given by fx (k) = (1 − p)k-¹p, k = = 1, 2,... (i) Verify (by calculation) that the log-likelihood function for the parameter p is given by l(x;p) = n (logp + (x − 1) log(1 − p)). - (ii) Derive the MLE estimator for p, and use it to compute the MLE estimate for the following sample data ¹: 1 3 1 1 1 2 3 1 1 1 1 (iii) What is the MLE estimator for 1/p? Verify this estimator is unbiased.

Consider a random variable V modeling the version of the software that people have installed on their computers, where the possible versions are {1, 2, 3, 4}. People are equally likely to have versions 1 and 2 and also equally likely to have versions 3 and 4. People are also x times more likely to have version 3 than version 2. (a) What is the (probability mass function) PMF of V as a function of x (b) What is the expected version of the software. Hint: the expected value of V as a function of x (c) What is the minimum value of x to ensure that the mean version is at least version 3?

You have two random variables M and N where the joint pdf is given by  f_M,N(m,n) = 1/2 if m + n >=2, m <=2 and n <=2 and 0 o/w. What is the cdf?