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What happened to the -1 in Gamma(alpha + n/2 , ...)? In other words, shouldn't it be Gamma(alpha - 1 + n/2, ...)?
Take a look at the Gamma density carefully. Recall we had Gamma(alpha, beta) to start with and the density has alpha-1 in it. That -1 gets carried over to the posterior too.
What happened to the -1 in Gamma(alpha + n/2 , ...)? In other words, shouldn't it be Gamma(alpha - 1 + n/2, ...)?
Take a look at the Gamma density carefully. Recall we had Gamma(alpha, beta) to start with and the density has alpha-1 in it. That -1 gets carried over to the posterior too.