where are the (possibly new) values of the characteristic in group . Equation (2) shows that: {| border="0" cellpadding="0" cellspacing="0" width="100%" |- | align="left" | | align="right" | |} Call the change in characteristic value from parent to child populations so that . As seen in Equation (1), the expected value operator is linear, so {| border="0" cellpadding="0" cellspacing="0" width="100%" |- | align="left" | | align="right" | |} Combining Equations (7) and (8) leads to {| border="0" cellpadding="0" cellspacing="0" width="100%" |- | align="left" | | align="right" | |} Now, let's compute the first term in the equality above. From Equation (1), we know that: Substituting the definition of fitness, (Equation (4)), we get: {| border="0" cellpadding="0" cellspacing="0" width="100%" |- | align="left" | | align="right" | |} Next, substituting the definitions of average fitness () from Equation (5), and average child characteristics () from Equation (6) gives the Price equation: Derivation of the continuous-time Price equation Consider a set of groups with that are characterized by a particular trait, denoted by . The number of individuals belonging to group experiences exponential growth:where corresponds to the fitness of the group. We want to derive an equation describing the time-evolution of the expected value of the trait:Based on the chain rule, we may derive an ordinary differential equation:A further application of the chain rule for gives us:Summing up the components gives us that: which is also known as the replicator equation. Now, note that: Therefore, putting all of these components together, we arrive at the continuous-time Price equation: Simple Price equation When the characteristic values do not change from the parent to the child generation, the second term in the Price equation becomes zero resulting in a simplified version of the Price equation: which can be restated as: where is the fractional fitness: . This simple Price equation can be proven using the definition in Equation (2) above. It makes this fundamental statement about evolution: "If a certain inheritable characteristic is correlated with an increase in fractional fitness, the average value of that characteristic in the child population will be increased over that in the parent population." Applications The Price equation can describe any system that changes over time, but is most often applied in evolutionary biology. The evolution of sight provides an example of simple directional selection. The evolution of sickle cell anemia shows how a heterozygote advantage can affect trait evolution.