Question

An emergency department research group is interested in developing a statistical model for the number of in-hospital deaths among patients presenting to the emergency department with heart failure. Let N be the number of patients presenting to the emergency department with heart failure in a given year and assume that this random variable follows a geometric distribution
p(N = n) = theta(1-theta)^n-1 n=1,...,infinite 0<theta<1

Next, for the ith patient define the random variable Yi to be equal to 1 if the patient dies during their hospital stay and Yi = 0 if they are discharged alive, withP(Yi = 1) =pi with 0<pi< 1. Assume that the patients and their mortality outcomes are all independent of each other. The random variable of interest to the researchers is T, the total number of deaths among the patients presenting with heart failure in a given year, and is defined as
T = Y1 + Y2 +...+ YN:
b)Find expression for Corr(N,T), in terms of pi and theta

c)Find an explicit expression for P(T=0)

Answer 1

Here, 7 be the total number of deaths among the pati ents presenting with heart failure in a given year, and is defined as That means, 7 is compound distributi on The random variable Y-- patient dies during their hospital sta 0, patients are discharged alive with P(Y -1)-p, and 0<P, <1 Thus, щ,-E(Y) The random variable V be the number of pati ents presenting to the emergency department with heart failure in a given year, and it foll ows geometric distribution, Therefore, the mean and variance of N are Therefore, the mean and Variance of T in terms of parameters θ andp, as follows 9p, (1- P)+P(1-e)

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