The likelihood function is the [[joint probability]] of the data given the hypothesis (i.e., the parameters). $L(\theta) = f(x|\theta)$ In simple terms, it is the probability of seeing the data given the hypothesis. Consider a coin toss, where the hypothesis is a fair coin (e.g., $p=0.5$). The probability of seeing five heads in a row is $0.5^5$. Thus the likelihood (from the [[Binomial distribution]]) is $L(p=0.5) = {n \choose k} p^k(1-p)^{n-k} = (0.5)^5 * (1-0.5)^0 = \frac{1}{32}$ For continuous random variables, the probability $P(X_1 = x_1, X_2 = x_2, \dots, X_n = x_n)$ can be expressed as the [[joint probability density function]]. $f(\vec{x}; \theta) = \prod_{i=1}^n f(x_i; \theta)$