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The likelihood function is not a probability distribution because its integral over the parameter space does not equal 1. It measures how well each parameter value explains the fixed data, rather than assigning probabilities to the parameter values themselves.

Conditions: The data are fixed.; The parameter $p$ is varied over its space $[0,1]$.

The likelihood function is not a probability distribution because its integral over the parameter space does not equal 1. It measures how well each parameter value explains the fixed data, rather than assigning probabilities to the parameter values themselves.

Conditions: The data are fixed.; The parameter $p$ is varied over its space $[0,1]$.

The likelihood function is a function of the unknown parameter $\theta$ after the observed data $x_1, \dots, x_n$ are fixed. It assigns to each candidate parameter value a measure of how likely the observed data are under that parameter.

Conditions: The data $x_1, \dots, x_n$ are treated as fixed observed constants.; The argument being varied is the parameter $\theta$.

The likelihood function is a function of the unknown parameter $\theta$ after the observed data $x_1, \dots, x_n$ are fixed. It assigns to each candidate parameter value a measure of how likely the observed data are under that parameter.

Conditions: The data $x_1, \dots, x_n$ are treated as fixed observed constants.; The argument being varied is the parameter $\theta$.

The bars represent the probabilities of all possible outcomes (head counts) for a fixed model. Because they exhaust the entire sample space, their sum must equal 1, which is the defining normalization property of a probability distribution.

Conditions: The coin model is fixed at $p=0.5$.; The number of flips is 10.; $k$ indexes all possible head counts from 0 to 10.

The bars represent the probabilities of all possible outcomes (head counts) for a fixed model. Because they exhaust the entire sample space, their sum must equal 1, which is the defining normalization property of a probability distribution.

Conditions: The coin model is fixed at $p=0.5$.; The number of flips is 10.; $k$ indexes all possible head counts from 0 to 10.