Possible error in Expectation for random sums

Poisson sums

For Poisson sums

Y=

N

∑

i=1

X

i

(i.e.

NPoisson(λ))

, we know that

Var(Y)=λE(

2

X

).

We also know that

Var(Y)=E(

2

Y

)-

2

E(Y)

.

I am not able to reconcile these two results using the Expectation function for

X

i

NormalDistribution(μ,σ),NPoisson(λ)

, as shown below:

Unexpected result from Mathematica

In[]:=

expectedResult=λ*Expectation[

2

X

,XNormalDistribution[μ,σ]];mathematicaResult=Expectation

2



N

∑

i=1

X

i



,{

X

i

NormalDistribution[μ,σ],NPoissonDistribution[λ]}-

2

Expectation[

N

∑

i=1

X

i

,{

X

i

NormalDistribution[μ,σ],NPoissonDistribution[λ]}]

;Simplify[expectedResult-mathematicaResult]

Out[]=

-

2

λ

2

σ

Conditional variance formula

Var(Y)=E(Var(Y|N))+Var(E(Y|N))

.