The probability distribution sums to 1 because it accounts for all possible data outcomes under a fixed model, satisfying the axioms of probability. The likelihood curve does not integrate to 1 because it measures relative support for different parameter values given fixed data, and is not constrained to be a probability distribution over the parameter space.
Conditions: Probability view: parameter fixed, data varies over the entire sample space.; Likelihood view: data fixed, parameter varies over its domain.
The probability distribution sums to 1 because it accounts for all possible data outcomes under a fixed model, satisfying the axioms of probability. The likelihood curve does not integrate to 1 because it measures relative support for different parameter values given fixed data, and is not constrained to be a probability distribution over the parameter space.
Conditions: Probability view: parameter fixed, data varies over the entire sample space.; Likelihood view: data fixed, parameter varies over its domain.