 Solution

Let $A\in \{0,1\}$ and $B\in \{0,1\}$ encode the alleles at two loci on a randomly chosen population haplotype. Write $p_{ab}=P(A=a,B=b)$, $p_A=p_{10}+p_{11}$ and $p_B=p_{01}+p_{11}$. The linkage disequilibrium coefficient isThe determinant identity follows by substituting the marginal frequencies and $\sum p_{ab}=1$. All four haplotype frequencies can be recovered asThus $D=0$ is precisely independence of the two allelic states. Its sign depends on which alleles are labeled one, and its possible magnitude depends on the marginal frequencies.

One other measure is squared allelic correlation. For segregating loci, $0<p_A,p_B<1$, their indicator-variable covariance is $D$, and their variances are $p_A(1-p_A)$ and $p_B(1-p_B)$. HenceThis is the squared correlation, useful for measuring how well one marker predicts the other in association mapping. It is undefined if a locus is monomorphic. Linkage disequilibrium describes population haplotype frequencies; genetic linkage describes transmission within meioses. Physically linked loci can have $D=0$, and population mixing can produce LD even between unlinked loci.