Hi there,
I'm doing an introductory statistics course and I'm having a little trouble with one of the questions. I am supposed to find out what sign the correlation coefficient is (ie, positive or negative) without calculating it. X is defined as the sum of N independent Bernoulli trials where probability of each success is p. N is a random variable that behaves according to a Poisson distribution with mean lamda.
Since N is an input to the computation of X, they should be correlated. I feel that by proving that the E(X|N=ni) increases as n increases I can conclude that the correlation coefficient between the X and N will be positive. Am I right in this assumption?
If so, how should I go about proving that E(X|N=ni) increases as N increases? I already calculated the conditional probability of X given a fixed value of N (which is essentially a binomial distribution) and also the joint probability. Will plugging in values (eg, N=1,2) and then calculating the E(X|N=1) and E(X|N=2) and then saying that the expected value when N=1 is higher than when N=2 be sufficient? Also is E(X|N=n)=pn (ie E(X|N=1)=p and E(X|N=2)=2p)?
Thanks for taking the time to read this (I hope I wasn't too confusing...)
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