At the first generation there is the group G0={e1,e2,e3} with e2 having the same genotype as e3. The maximal number of child is 2. The distribution for e1 in all groups is given with d0=1/4 d1=3/4 d2=0. The distribution for e2 and e3 is for all groups but {e1,e2} 1/2 1/3 1/6. For the group {e1,e2} it is given with 1/7 4/7 2/7.
What is the probability of the population {e2} in the second generation?
Because e3 and e2 have the same genotype e2 is equated with e3.
This 15 groups that can come frome G0:
{},{e2},{e2,e2},{e2,e2,e2},{e2,e2,e2,e2},
{e1},{e1,e2},{e1,e2,e2},{e1,e2,e2,e2},{e1,e2,e2,e2,e2}
{e1,e1},{e1,e1,e2},{e1,e1,e2,e2},{e1,e1,e2,e2,e2},{e1,e1,e2,e2,e2,e2}
{e1,e2,e2}->{}->{e2} | ..*0 | 0 |
{e1,e2,e2}->{e2}->{e2} | ( (1/4*1/2*1/3)*2)*1/3 | 1/36 |
{e1,e2,e2}->{e2,e2}->{e2} | ( (1/4*1/2*1/6)*2+(1/4*1/3*1/3))*1/2*1/3*2 | 5/216 |
{e1,e2,e2}->{e2,e2,e2}->{e2} | ( (1/4*1/3*1/6)*2)*1/2*1/2*1/3*3 | 1/144 |
{e1,e2,e2}->{e2,e2,e2,e2}->{e2} | (1/4*1/6*1/6)*1/2*1/2*1/2*1/3*4 | 1/864 |
{e1,e2,e2}->{e1}->{e2} | ..*0 | 0 |
{e1,e2,e2}->{e1,e2}->{e2} | ( (3/4*1/2*1/3)*2)*1/4*4/7 | 1/28 |
{e1,e2,e2}->{e1,e2,e2}->{e2} | ( (3/4*1/2*1/6)*2+(3/4*1/3*1/3))*1/4*1/2*1/3*2 | 5/288 |
{e1,e2,e2}->{e1,e2,e2,e2}->{e2} | ( (3/4*1/3*1/6)*2)*1/4*1/2*1/2*1/3*3 | 1/192 |
{e1,e2,e2}->{e1,e2,e2,e2,e2}->{e2} | ( (3/4*1/6*1/6)*1/4*1/2*1/2*1/2*1/3*4 | 1/1152 |
{e1,e2,e2}->{e1,e1}->{e2} | 0*0 | 0 |
{e1,e2,e2}->{e1,e1,e2}->{e2} | 0*.. | 0 |
{e1,e2,e2}->{e1,e1,e2,e2}->{e2} | 0*.. | 0 |
{e1,e2,e2}->{e1,e1,e2,e2,e2}->{e2} | 0*.. | 0 |
{e1,e2,e2}->{e1,e1,e2,e2,e2,e2}->{e2} | 0*.. | 0 |
((1/4)*(1/2*1/2))*((1)*(0)) | 0 |
+((1/4)*(1/2*1/3+1/3*1/2))*((1)*(1/3)) | 1/36 |
+((1/4)*(1/2*1/6+1/3*1/3+1/6*1/2))*((1)*(1/2*1/3+1/3*1/2)) | 5/216 |
+((1/4)*(1/3*1/6+1/6*1/3))*((1)*(1/2*1/2*1/3+1/2*1/3*1/2+1/3*1/2*1/2)) |
1/144 |
+((1/4)*(1/6*1/6))*((1)*(1/2*1/2*1/2*1/3+1/2*1/2*1/3*1/2+1/2*1/3*1/2*1/2+1/3*1/2*1/2*1/2)) |
1/864 |
+((3/4)*(1/2*1/2))*((1/4)*(0)) | 0 |
+((3/4)*(1/2*1/3+1/3*1/2))*((1/4)*(4/7)) | 1/28 |
+((3/4)*(1/2*1/6+1/3*1/3+1/6*1/2))*((1/4)*(1/2*1/3+1/3*1/2)) | 5/288 |
+((3/4)*(1/3*1/6+1/6*1/3))*((1/4)*(1/2*1/2*1/3+1/2*1/3*1/2+1/3*1/2*1/2)) | 1/192 |
+((3/4)*(1/6*1/6))*((1/4)*(1/2*1/2*1/2*1/3+1/2*1/2*1/3*1/2+1/2*1/3*1/2*1/2+1/3*1/2*1/2*1/2)) | 1/1152 |
+((0)*(1/2*1/2))*((1/4*1/4)*(0)) | 0 |
+((0)*(1/2*1/3+1/3*1/2))*((1/4*1/4)*(1/3)) | 0 |
+((0)*(1/2*1/6+1/3*1/3+1/6*1/3))*((1/4*1/4)*(1/2*1/3+1/3*1/2)) | 0 |
+((0)*(1/3*1/6+1/6*1/3))*((1/4*1/4)*(1/2*1/2*1/3+1/2*1/3*1/2+1/3*1/2*1/2)) | 0 |
+((0)*(1/6*1/6))*((1/4*1/4)*(1/2*1/2*1/2*1/3+1/2*1/2*1/3*1/2+1/2*1/3*1/2*1/2+1/3*1/2*1/2*1/2)) | 0 |
wkat1kbt2 = S[k0,k1,..,kt2-t1 : k0=ka; k2..kt2-t1-1 >= 0; kt2-t1=kb]P[t :
0<=t<t2-t1]P[j]vjktkt+1t1+t
=S[k0,k1,..,kt2-t1 : k0=ka; k2..kt2-t1-1 >= 0; kt2-t1=kb]P[t :
0<=t<t2-t1]P[j]S[p1,..,px : x=N(e'jkt)]
1(y=N(e'jkt+1),p1,..,px)*dp1ijktt*..*dpxijktt
Copyright ©2008 Henri Steyer