You are interested in studying the writing style of a popular Time Magazine contributor, FZ. You collect a simple random sample of his articles and count how many times he uses the word "however" in each of the articles in your sample, (x1,,xn)(x1,,xn). You also count the length of each article in your sample, (y1,,yn). In this set-up, xi is the number of times the word "however" appeared in the i-th article and yi is the total number of words in the i-th article.

Let Xi denote the quantity that captures the number of times the word "however" appears in the i-th article. Let's assume that the quantities X1,,Xn are independent and identically distributed (IID) according to a Poisson distribution with unknown parameter v.1000yi

p(Xi=xiyi,v,1000) = Poisson(xiv.1000yi) for i=1,,n.

• Define the likelihood L(v)=p(..) for this model and set-up at the highest level of abstraction.
• Write the likelihood L(v) for a generic sample of n articles, (xi,,xn), and n article lengths, (y1,,yn) .
• Write the log-likelihood l(v) for a generic sample of n articles, (xi,,xn), and n article lengths, (y1,,yn).