Likelihood function
Discrete probability distribution

Let X {\textstyle X} be a discrete random variable with probability mass function p {\textstyle p} depending on a parameter θ {\textstyle \theta }. Then the function

L (θ ∣ x) = p θ (x) = P θ (X = x), {\displaystyle {\mathcal {L}}(\theta \mid x)=p_{\theta }(x)=P_{\theta }(X=x),}

considered as a function of θ {\textstyle \theta }, is the likelihood function, given the outcome x {\textstyle x} of the random variable X {\textstyle X}. Sometimes the probability of "the value x {\textstyle x} of X {\textstyle X} for the parameter value θ {\textstyle \theta } " is written as P (X = x | θ) or P (X = x; θ). The likelihood is the probability that a particular outcome x {\textstyle x} is observed when the true value of the parameter is θ {\textstyle \theta }, equivalent to the probability mass on x {\textstyle x}; it is not a probability density over the parameter θ {\textstyle \theta }. The likelihood, L (θ ∣ x) {\textstyle {\mathcal {L}}(\theta \mid x)}, should not be confused with P (θ ∣ x) {\textstyle P(\theta \mid x)}, which is the posterior probability of θ {\textstyle \theta } given the data x {\textstyle x}.
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