Non-central chi-squared distribution, scaled or not?

Question

I am a bit confused as to the non-central chi-squared scaling... It may sound like a basic question to most, not for me!

Suppose we have a variable (for the sake of argument, that has only one degree of freedom) that is defined as

$Y = \lambda + (\sigma X + \mu)^2$, with $X\sim\mathcal{N}(0,1)$.

We have $(\sigma X + \mu)^2 \sim \mathcal{X'}^2_1(\mu^2/\sigma^2)$ and thus, we can say

$(Y - \lambda) \sim \mathcal{X'}^2_1(\mu^2/\sigma^2)$.

Alternatively, we can reformulate the variable as

$Y = \lambda + \sigma^2(X + \mu/\sigma)^2$, with $X\sim\mathcal{N}(0,1)$.

We have $(X + \mu/\sigma)^2 \sim \mathcal{X'}^2_1(\mu ^2/\sigma^2)$ and thus, we can say

$(Y - \lambda)/\sigma^2 \sim \mathcal{X'}^2_1(\mu^2/\sigma^2)$.

Which one is correct?

Answer 1

Let $Y=\lambda+(\sigma X+\mu)^2$. We have

$$\begin{split}X&\sim N(0,1)\\ \sigma X&\sim N(0, \sigma^2)\\ \sigma X+\mu&\sim N(\mu, \sigma)^2\text{ (*)}\\ \left(\frac{\sigma X+\mu}{\sigma}\right)&\sim N\left(\frac \mu \sigma, 1\right)\\ \left(\frac{\sigma X+\mu}{\sigma}\right)^2&\sim\chi^2_{1, \frac{\mu^2}{\sigma^2}}\end{split}$$

Therefore

$$\frac{Y-\lambda}{\sigma^2}=\left(\frac{\sigma X+\mu}{\sigma}\right)^2\sim\chi^2_{1, \frac{\mu^2}{\sigma^2}}$$

the second is correct. Notice that $\sigma X+\mu$ does not have unit variance (*) so squaring it does not create a non-central chi squared random variable, which requires it be the sum of independent, normally distributed variables with unit variance. Therefore the first is not correct.

Answer 2

The second expression is correct, the first is incorrect since

$$ \sigma X + \mu \sim N(\mu, \sigma^2) $$

and squaring this random variable does not give a non-central chi-square - by definition a non-central chisq is the square of a normal random variable with mean $\mu$ and variance $1$.

It follows that

$$ \frac{Y-\lambda}{\sigma^2} \sim \chi^2_1(\mu^2/\sigma^2) $$