TITLE: Poisson Distribution with Conditional Expectation
QUESTION [0 upvotes]: my friend asked me a question on poisson distribution but I don't know how to do it. 
Your friend sometimes eats insomnia cookies before bed. The number of cookies she eats is a Poisson random variable, C, with λ = 2. The no. of hours she sleeps is a normal random variable, µ = 7, σ = 1. When she eats more cookies, she sleeps less on average.
a) Is covariance of S and C positive or negative? 
b) Is E[S|C = 3] greater than seven or less than seven?
c) Is E[C|S = 6] greater than 2 or less than 2
a) I got it as negative
b) I got it as less than seven but I don't know how to prove it
c) I got it as greater than 2 but I don't know how to prove it too.
Is there a way to prove conditional expectation?

REPLY [0 votes]: Start by finding the sign of the covariance. It should be pretty intuitive. Eating more cookies implies sleeping less. The covariance is the way of measuring how one variables varies with another, it maybe helpful to think it always the same sign as the correlation coefficient.
Then for b) and c) think about it this way, am I eating more or less than the average? Based on the covariance you just found, should my conditioned variable be above or below the average?
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