Drift 2

3I.B The Stationary DistributionA different way of thinking of genetic drift rather than the probability of a singlepopulation having a particular state is as the probability distribution of a large number ofreplicate populations with similar properties.Let X i be a random variable that denotes the frequency of populations having i copies ofthe A1 allele (i.e., being in state i, i = 0, 1, 2, ... 2N)Let X be the ROW vector of X i , i.e., [X 0 , X 1 , X 2 , ... X N ]After one generation of genetic drift, X' = XT2NX ii=0where Xj ' = ∑T ij-------------------------------------------------------------------------------------------------------NOTES on matrix multiplication (from H. Anton, "Elementary Linear Algebra")Definition: If A is an m x r matrix and B is an r x n matrix, then the PRODUCT ABis the m x n matrix whose entries are determined as follows. To find the entry inrow i and column j of AB, single out row i from the matrix A and the column j fromthe matrix B. Multiply the corresponding entries from the row and columntogether and then add up the resulting products.For example:A = 1 2 4 and B = 4 1 4 32 6 0 0 -1 3 12 7 5 2Since A is a 2x3 matrix and B is a 3x4 matrix, the product AB is a 2x4 matrix. Todetermine, say, the entry in row 2, column 3 of AB, single out row 2 from A and column3 from B. Then, multiply corresponding entries together and add up these products.1 2 4 4 1 4 3 x x x x2 6 0 0 -1 3 1 = x x 26 x2 7 5 2For more in-depth information on matrix multiplication, check out the following websites.http://mathworld.wolfram.com/MatrixMultiplication.htmlhttp://www.mai.liu.se/~halun/matrix/matrix.html