Prove the assertion made in the text after Definition 8 2 1 If f(x ) is the pmf of a discrete random variable, then the numerator of (x), the LRT statistic, is the maximum probability of the observed sample when the maximum is computed over parameters in the null hypothesis Furthermore, the denomin...

For discrete random variables ...

Prove the assertion made in the text-after Definition 8.2.1. If f(x|) is the pmf of a discrete

Question:

Prove the assertion made in the text-after Definition 8.2.1. If f(x|θ) is the pmf of a discrete random variable, then the numerator of λ(x), the LRT statistic, is the maximum probability of the observed sample when the maximum is computed over parameters in the null hypothesis. Furthermore, the denominator of A(x) is the maximum probability of the observed sample over all possible parameters.