= Solution
\b[The prediction depends on whether effect means a phenotypic change or a <genotype fitness> advantage, and whether we sample new <mutations>, successful substitutions at one stage, or a complete adaptive walk.] These distributions differ even in simple models. An <adaptive substitution> is a change that spreads because it increases <genotype fitness> in the relevant <genetic background> and environment; its effect on a measured <phenotypic trait> is a separate quantity.
<Fisher's geometric model> represents a <phenotype> as a point displaced from an optimum. As derived in question 3, a <mutation> is favorable when $2\mathbf z\mathbin{\cdot}\mathbf a+\|\mathbf a\|^2<0$. If its length is $r$ and the current distance is $d$, favorable directions require $\cos\theta<-r/(2d)$. Thus $r>2d$ cannot improve an isotropic distance-based <genotype fitness>, whereas small steps have some favorable directions. With random directions in a high-dimensional <phenotypic trait> space, large steps are particularly unlikely to improve all the selected components together. This is a probability-of-benefit argument, not yet a prediction for what fixes: extremely small advantageous effects are easily lost through <genetic drift>.
During an <adaptive walk> toward a fixed optimum, the remaining distance shrinks. Substantial changes can be favorable early, whereas fine adjustments dominate late. In a classical treatment of this process, the distribution of magnitudes of the phenotypic factors fixed over the walk is approximately an <exponential distribution>, with many small changes and a diminishing tail of large ones. This can reconcile a few prominent changes with numerous minor contributors; it is not the assertion that every step must be tiny.<The population genetics of adaptation: the distribution of factors fixed during adaptive evolution by H. Allen Orr>[Evolution 52, pp. 935–949; approximate distribution of factors fixed during approach to a phenotypic optimum]{ref} The shape concerns the stated model and effect measure. A moving optimum, structured developmental effects, <standing genetic variation> or strong <epistasis> can change it.
A related but distinct argument uses <extreme value theory>. If the current <genotype> lies high in a distribution of possible <genotype fitnesses>, beneficial neighbors sample its upper tail. For a broad class of rapidly declining tails, excesses above a high threshold approximately follow an <exponential distribution>. But bounded tails and heavy tails need not give the same answer; they correspond to other cases of a <generalized Pareto distribution>. Neither an exponential tail nor the assumption of a high initial threshold is a law of nature. Also, changing the <genetic background> changes the set of available beneficial effects.
Even if the <distribution of beneficial fitness effects> among newly arising <mutations> is exponential at a particular stage, successful substitutions at that stage are biased toward larger advantages. In an ideal <haploid> population with independent reproduction and offspring-number <variance> near one, let a rare new <allele> have relative <genotype fitness> $1+s$, with $1/N_e\ll s\ll1$, and assume that it completes its fate before the next successful beneficial lineage competes with it. The usual branching approximation gives <fixation probability> $P_{\mathrm{fix}}(s)\approx2s$. More generally, if $f_b(s)$ includes the relative arrival rates of beneficial <mutations>,
$$
g_{\mathrm{fix}}(s)=\frac{f_b(s)P_{\mathrm{fix}}(s)}{\int_0^\infty f_b(u)P_{\mathrm{fix}}(u)\,du}.
$$
For the illustrative small-effect approximation $f_b(s)=\lambda^{-1}e^{-s/\lambda}$, this becomes
$$
\boxed{g_{\mathrm{fix}}(s)\approx\frac{s}{\lambda^2}e^{-s/\lambda},\qquad
\mathbb E_{\mathrm{fix}}[s]\approx2\lambda.}
$$
It is a <gamma distribution> of shape 2, not an <exponential distribution>. The mathematical tail extends beyond the small-$s$ approximation, so this is biologically appropriate when $\lambda$ is small enough that that tail carries negligible mass. With $\lambda=0.01$, the fraction exceeding $s=0.02$ is $e^{-2}\approx0.135$ among arrivals but $3e^{-2}\approx0.406$ among substitutions in this approximation. This calculation and the pooled phenotypic-walk result answer different questions and are not contradictory.
Tests can begin with <experimental evolution>. Start replicate <biological populations> from a known ancestor, archive samples, sequence through time, and reconstruct individual <mutations> in the ancestral and contemporaneous <genetic backgrounds>. Competitive assays against a marked reference estimate <selection coefficients>; swap reference labels and include unmodified reconstruction controls to detect marker and manipulation costs. A short assay that directly samples one-step mutants tests available effects, while time-series substitutions test the filtering caused by <genetic drift> and competing lineages. In particular, experiments on two viruses rejected an exponential model for the beneficial effects they sampled, showing why tail shape must be tested rather than assumed.<Beneficial fitness effects are not exponential for two viruses by Darin R. Rokyta and others>[Journal of Molecular Evolution 67, pp. 368–376; inferred upper-tail shape for the two viral systems]{ref}
Natural <adaptation> can be studied with <quantitative trait locus mapping>, <association mapping> and functional reconstruction. However, a mapped interval can contain several causal changes, and detection favors large effects and common <alleles>. Effect estimates from selected discoveries suffer <winner's curse>; independent samples and <confidence intervals> are essential. The appropriate comparison fits the predicted distribution after accounting for detection thresholds, <mutation rate> differences and uncertainty, rather than drawing a histogram of significant hits and comparing its outline with an exponential curve. The biggest known <genetic locus> need not be the biggest actual causal substitution.
Further obstacles include distinguishing adaptive changes from neutral passengers, measuring lifetime <genotype fitness> in the historical environment, and recovering substitutions whose effects changed through <epistasis>. <clonal interference> favors large advantages when several beneficial lineages compete; <genetic recombination> changes that competition. <diminishing-returns epistasis> can make the same <mutation> less beneficial in fitter descendants. Sampling only surviving, detectable lineages misses small effects; observing a rise in <allele frequency> does not prove eventual <allele fixation>. \b[Theory offers conditional distributional predictions that are testable only when the effect measure, sampling stage, population regime and observation process are matched.]
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