Can you explain why Y=k that doesn’t make sense with the Y=N-X equation. Couldn’

Question

Can you explain why Y=k that doesn’t make sense with the Y=N-X equation. Couldn’t you just sub k for N-Y? . Binomial prools Let X denote a bincenial andom vciale with paranetees (N, p). (a) Show that Y-N-X is a binominl neloen variable with pa (N, 1-p 2016 (b) what b P(x nn)? Hint : Uwthe binomial thronnan to write arl expresion for {r+p)" +(z-y)" and then met z-1-py"p. If you are not familiar with the binomial theoresm, it should be easy to look up Solution )We have that Y=N-X X=N-Y,so that N-A where q 1-p. We also hnve that Therefore so that fnd that Y has a binomial distribation with parame tees (N,g)-(N.1 -p) b) We hawe Using the hint, we iest dusing the Binomial thhoorem that and (-, + (1-p))N (2)(-no-nN - Adding these two esqpressions, we hae Sce1-1) is aero for & odd, and 2 for & even, we fn that

Explanation / Answer

ans) You should not substitute k for (N-Y) in the equation Y=N-X because Y itself is a random variable.For a particular value of X the probability mass function should be a single value which represents the mass at that value. If you substitute k for (N-Y) then for a particular value of X the probalility mass function itself will be a random variable. Moreover you have to find probability mass function of Y for a particular value. For a particular value of X we know the pmf of X. So we find the pmf of Y using the pmf of X.

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