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<Task/>
<Group labelreference="L1" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">restart:
assume(alpha_n&gt;0,beta_n&gt;0,b_n&gt;0,b_nM1&gt;0);
with(Statistics):
pdf:=PDF(RandomVariable(Normal(0,1)),x):
P:=x:
int(x*P*P*pdf,x=-infinity..infinity);
int(x*P*x*P*pdf,x=-infinity..infinity);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output">IiIh</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output">IiIk</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L2" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">res:=collect(algsubs(P_n*P_nM1*t=P_n^2,expand(((t-alpha_n)*P_n-beta_n*P_nM1)^2)),t);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">LCwqJkkkUF9uRzYiIiIjSSJ0R0YlRiYiIiIqKEYkRiZGJ0YoSSlhbHBoYV9ufGlyRzYjL0knX2xvY2FsRyUqcHJvdGVjdGVkR0kyX24xNDA1NjA1MTMxMDE0MDhHRiVGKCEiIyomLCYqJEYqRiZGKEkoYmV0YV9ufGlyRzYjL0YtSTJfbjE0MDU2MDUxMzA1MTc3NkdGJUYwRihGJEYmRigqKkYkRihJJlBfbk0xR0YlRihGNEYoRipGKEYmKiZGOUYmRjRGJkYo</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L3" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">res1:=algsubs(P_n*P_nM1=sqrt(b_n*b_nM1)*pi_n*pi_nM1,subs(P_n^2=b_n*pi_n^2,P_nM1^2=b_nM1*pi_nM1^2,res)):
K0:=coeff(res1,t,0):
res1:=res1-K0+algsubs(pi_n*pi_nM1=0,subs(pi_n^2=1,pi_nM1^2=1,K0));
K1:=coeff(res1,t,1):
res1:=collect(res1-K1*t+algsubs(pi_n^2*t=alpha_n,expand(K1*t)),t);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">LCwqKEklYl9ufGlyRzYjL0knX2xvY2FsRyUqcHJvdGVjdGVkR0kyX24xNDA1NjA1MTMwNTExMDRHNiIiIiJJJXBpX25HRioiIiNJInRHRipGLUYrKipGJEYrRixGLUYuRitJKWFscGhhX258aXJHNiMvRidJMl9uMTQwNTYwNTEzMTAxNDA4R0YqRishIiMqJkknYl9uTTF8aXJHNiMvRidJMl9uMTQwNTYwNTEzMDUxNDI0R0YqRitJKGJldGFfbnxpckc2Iy9GJ0kyX24xNDA1NjA1MTMwNTE3NzZHRipGLUYrKiZGJEYrRjBGLUYrKiZGJEYrRjpGK0Y0</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">LCoqKEklYl9ufGlyRzYjL0knX2xvY2FsRyUqcHJvdGVjdGVkR0kyX24xNDA1NjA1MTMwNTExMDRHNiIiIiJJJXBpX25HRioiIiNJInRHRipGLUYrKiZJJ2Jfbk0xfGlyRzYjL0YnSTJfbjE0MDU2MDUxMzA1MTQyNEdGKkYrSShiZXRhX258aXJHNiMvRidJMl9uMTQwNTYwNTEzMDUxNzc2R0YqRi1GKyomRiRGK0kpYWxwaGFfbnxpckc2Iy9GJ0kyX24xNDA1NjA1MTMxMDE0MDhHRipGLSEiIiomRiRGK0Y0RishIiM=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L4" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">res2:=simplify(subs(b_nM1=b_n/beta_n,expand(simplify(res1/b_n))));</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">LCgqJkklcGlfbkc2IiIiI0kidEdGJUYmIiIiKiRJKWFscGhhX258aXJHNiMvSSdfbG9jYWxHJSpwcm90ZWN0ZWRHSTJfbjE0MDU2MDUxMzEwMTQwOEdGJUYmISIiSShiZXRhX258aXJHNiMvRi1JMl9uMTQwNTYwNTEzMDUxNzc2R0YlRjA=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L5" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal"></Text-field>
</Input>
</Group>
</Worksheet>