# Analytical Expression for Moment of Generalized chi-squared distribution

Consider $$Z\sim N(0,1)$$ and the moments: $$E\left[ \left(C_1(Z+\sqrt{\lambda})^2 - C_2\right)^t\right].$$ Here, $$C_1$$, $$\lambda$$ and $$C_2$$ are arbitrary constants which are all positive. $$t$$ is a nonnegative integer. Is there an analytical form for these moments in terms of $$t$$, $$\lambda$$, $$C_1$$ and $$C_2$$?

So far, I have identified that $$C_1(A+\sqrt{\lambda})^2 + C_2$$ follows a generalized chi-squared distribution. Where, following the notation on the wiki, $$k=1$$, $$\lambda = \lambda$$, $$w = C_1$$, $$m=C_2$$, $$s=0$$. However, I do not see any analytical expression for the moments on the wiki.

I see there are analytical moments for the noncentral chi-squared distribution. I also know how the moment generating function of the noncentral chi-squared distribution can be used to obtain moments for the generalized chi-squared distribution. However, I am able to obtain an analytical expressions using that approach.

• I imagine lambda is positive, is t as well, or either C for that matter? Jul 29 at 19:27
• For what it's worth, Mathematica could not solve this integral, even with the assumptions that t and lambda > 0. (Edit: Integral t doesn't help either, unfortunately). Jul 29 at 19:29
• Assuming $t$ is a non-negative integer, expand the argument as a polynomial in $Z$ and apply linearity of expectation.
– whuber
Jul 29 at 20:03
• Thank you for that last comment. I will post an answer below. I feel that this is the correct method. Jul 29 at 20:49

Following the suggestion provided by @whuber: $$\left(C_1(A + \sqrt{\lambda})^2-C_2\right)^t = \sum_{k=0}^t\text{nCr}(t, k)\left(C_1(A + \sqrt{\lambda})^2\right)^k\left(-C_2\right)^{t-k}$$ Then apply the expectation operator: $$\text{E}\left[\left(C_1(A + \sqrt{\lambda})^2+C_2\right)^t\right] = \sum_{k=0}^t\text{nCr}(t, k)C_1^k\text{E}\left[\left((A + \sqrt{\lambda})^2\right)^k\right]\left(-C_2\right)^{t-k}$$
Note that $$\text{E}\left[\left((A + \sqrt{\lambda})^2\right)^k\right]$$ is the $$k$$-th moment of a noncentral chi-squared distribution with analytical form reported here.