I need to refresh my memory on Probability Theory, especially on
conditional probability. In particular, I want to solve the following
two problems. Can somebody point me some good books on Probability
Theory? Thank you!
1. Z=X+Y, where X and Y are independent random variables and their
distributions are known.
Now, I want to compute E(X | Z = z).
2.Suppose that I have $I \times J$ random number in I by J cells. For
the random number in the cell on the i'th row and the j's column, it
follows Poisson distribution with the parameter $\mu_{ij}$.
I want to compute P(n_{i1},n_{i2},...,n_{iJ} | \sum_{j=1}^J n_{ij}),
which the probability distribution in a row conditioned on the row
sum.
Some book directly states that the conditional distribution is a
multinomial distribution with parameters (p_{i1},p_{i2},...,p_{iJ}),
where p_{ij} = \mu_{ij}/\sum_{j=1}^J \mu_{ij}. But I'm not sure how to
derive it.
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