In[1]:=

q=.

zbar = p^2 a + 2 p q d + q^2 (-a) ;

In[3]:=

α1 = p a + q d - zbar ;

α2 = q (-a) + p d - zbar ;

In[5]:=

FullSimplify[α1]

FullSimplify[α2]

Out[5]=

d (1 - 2 p) q + a (p - p^2 + q^2)

Out[6]=

d (p - 2 p q) - a (p^2 + q - q^2)

In[7]:=

BV22 = 2 α2 ;

BV12 = α2 + α1 ;

BV11 = 2α1 ;

In[10]:=

meanBV = p p BV11 + 2 p q BV12 + q q BV22 ;

In[11]:=

VarBV = p^2 (BV11 - meanBV)^2 + 2 p q (BV12 - meanBV)^2 + q^2 (BV22 - meanBV)^2 ;

In[12]:=

q = 1 - p ;

FullSimplify[VarBV]

Out[13]=

-2 (-1 + p) p (a + d - 2 d p)^2

We can rearrange VarBV to

VarBV = -2 (-q) p (a + d (1 - 2p))^2

since (1 - 2p) = (q - p) we get

-2 (-q) p (a + d (q - p))^2 hence

VarBV = 2 p q (a + d (q - p))^2

Since the variance in breeding values = additive genetic variance, the additive genetic variance

varA = varBV = 2 p q (a + d (q - p))^2

Now the mean breeding value should be zero .   To check this we simplify meanBV

In[14]:=

FullSimplify[meanBV]

Out[14]=

0

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