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The maximum likelihood estimate is defined as the parameter value that corresponds to the peak (maximum) of the likelihood curve. It is the value of the parameter that gives the highest likelihood for the observed data.

Conditions: A likelihood function $L(p)$ over candidate parameter values is available.; The data are fixed.

The maximum likelihood estimate is defined as the parameter value that corresponds to the peak (maximum) of the likelihood curve. It is the value of the parameter that gives the highest likelihood for the observed data.

Conditions: A likelihood function $L(p)$ over candidate parameter values is available.; The data are fixed.

The likelihood function is $L(p) = \binom{10}{7} \cdot p^7 \cdot (1-p)^3$. It represents the probability of observing 7 heads and 3 tails as a function of the unknown parameter $p$, the probability of heads.

Conditions: The number of tosses $n=10$.; The observed number of heads $k=7$.; $p$ is the probability of heads on a single toss.

The likelihood function is $L(p) = \binom{10}{7} \cdot p^7 \cdot (1-p)^3$. It represents the probability of observing 7 heads and 3 tails as a function of the unknown parameter $p$, the probability of heads.

Conditions: The number of tosses $n=10$.; The observed number of heads $k=7$.; $p$ is the probability of heads on a single toss.

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 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.

Among the tested parameter values, $p=0.7$ yields the highest likelihood for the observed data. The likelihood measures how well each candidate value of $p$ explains the fixed observation of 7 heads in 10 flips.

Conditions: The observed data are fixed at 7 heads in 10 flips.; The candidate values tested are $p=0.3, 0.5,$ and $0.7$.

Among the tested parameter values, $p=0.7$ yields the highest likelihood for the observed data. The likelihood measures how well each candidate value of $p$ explains the fixed observation of 7 heads in 10 flips.

Conditions: The observed data are fixed at 7 heads in 10 flips.; The candidate values tested are $p=0.3, 0.5,$ and $0.7$.