conditional probability - why can\'t we just cancel out the denominator and nume

Question

conditional probability - why can't we just cancel out the denominator and numerator? My question is written on the picture:

EXAMPLE 4b If X and Y are independent Poisson random variables with respective parameters 1 and 42, calculate the conditional distribution of X given that X + Y = n. Solution. We calculate the conditional probability mass function of X given that X -+ Y n as follows: n P(X + Y = n} Y = n} P(Y=n-k) both on the denominator and numerator?? PK + Y = n} P(X| + here, why can't we write P(Y=n-k) on the denominator and cancel out = where the last equality follows from the assumed independence of X and Y. Recalling (Example 3e) that X + Y has a Poisson distribution with parameter 1 + 2, we see that the preceding equals (n- k)! k! (1 + 2)" In other words, the conditional distribution of X given that X + Y distribution with parameters n and 1/A1 + 2) n is the binomial

Explanation / Answer

Good question. See k is some constant value it is not variable, constant as in, denominator considers all the values of k, while numerator takes only one of them. See it like this way P(X+Y=5) and P(X=2)*P(Y=3) are different. Hope it clears your doubt. :)

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