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If X ~ Bin (n, p) and Y ~ Bin (n, 1 - p). Verify that...

If X ~ Bin (n, p) and Y ~ Bin (n, 1 - p). Verify that for any k = 1, 2,....
P (X = k) = P (Y = n - k)

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Answer #1

, X has Binomial distribution with parameters, number of trials =n and success probability p.

The probability of X=k is

, Y has Binomial distribution with parameters, number of trials =n and success probability 1-p.

The probability of Y=y is

Hence, the probability of Y=n-k is

this is nothing but P(X=k)

Hence P (X = k) = P (Y = n - k) for any k=0,1,2,...,n

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