Slide 1: Understanding Expected Value and Variance - 🎯 Goal: Build intuition for how selection changes average fitness. - Expected value (mean fitness) - Formula: \(E[w] = \bar{w} = \sum_i p_i w_i\) - Each genotype’s fitness \(w_i\) is weighted by its frequency \(p_i\). - Expected value of squared fitness - Formula: \(E[w^2] = \sum_i p_i w_i^2\) - Used to compute how much individual fitnesses differ from the mean. - πŸ’‘ The gap between these two quantities (E[wΒ²] and (E[w])Β²) measures variance β€” the engine of natural selection.

Slide 2: Variance in Fitness (Concept + Formula) - πŸ“˜ Definition: \(\text{Var}(w) = E[w^2] - (E[w])^2\) - Expanded: \(\text{Var}(w) = \sum_i p_i w_i^2 - (\sum_i p_i w_i)^2\) - πŸ“Š Intuitive meaning: - High variance β†’ strong selection potential. - Zero variance β†’ no change in mean fitness. - πŸ” Analogy: Variance is like β€œfuel.” The more differences in fitness, the more power selection has to increase \(\bar{w}\).

Slide 3: Deriving Ξ”wΜ„ = Var(w)/wΜ„ (Visual Flow) 1️⃣ After selection: \(\bar{w}' = \sum_i p_i' w_i = \frac{\sum_i p_i w_i^2}{\bar{w}}\) 2️⃣ Subtract initial mean: \(\Delta \bar{w} = \bar{w}' - \bar{w} = \frac{1}{\bar{w}} \sum_i p_i w_i^2 - \bar{w}\) 3️⃣ Simplify using variance: \[\sum_i p_i w_i^2 - \bar{w}^2 = \text{Var}(w)\] 4️⃣ Final result: \[\boxed{\Delta \bar{w} = \frac{\text{Var}(w)}{\bar{w}}}\]

🧭 Interpretation: - The increase in mean fitness equals the variance in fitness scaled by the current mean. - Selection always increases mean fitness (if variance > 0).

Slide 4: Two-Genotype Example (Visualized) | Genotype | Frequency (\(p_i\)) | Fitness (\(w_i\)) | Contribution (\(p_i w_i\)) | |———–|——————–|β€”β€”β€”β€”β€”β€”|β€”β€”β€”β€”β€”β€”β€”β€”β€”-| | A | 0.5 | 1.0 | 0.5 | | B | 0.5 | 3.0 | 1.5 |

➑️ Compute: - \(E[w] = 2.0\) - \(E[w^2] = 5.0\) - \(\text{Var}(w) = 5 - 4 = 1\) - \(\Delta \bar{w} = 1 / 2 = 0.5\)

πŸ“ˆ Result: Mean fitness increases from 2.0 β†’ 2.5 in one generation. 🧬 Reason: B’s higher fitness raises the population average through selection.

Slide 5: Concept Summary (Stylized) 🌱 Core idea: Natural selection increases mean fitness in proportion to how much heritable variation in fitness exists.

🧩 Mathematical identity: \(\Delta \bar{w} = \frac{\text{Var}(w)}{\bar{w}}\)

βš™οΈ Biological interpretation: - Variance = β€œraw material” for selection. - Greater fitness spread β†’ faster adaptation.

πŸ“˜ Link to Fisher’s Fundamental Theorem: - Fisher extended this discrete rule to continuous time. - States: Rate of increase in mean Malthusian fitness (ln w) = additive genetic variance in fitness.

🎨 Design tips: - Use arrows to show algebra flow. - Use icons for mean (βš–οΈ), variance (πŸ“Š), and increase (πŸ“ˆ). - Highlight final formula in a colored box for emphasis.