what are the characteristics of this random variable?

Question

Consider independent sequences of independent and identically distributed (i.i.d.) zero-mean Gaussian random scalars $r_{i}\sim\mathcal{N}(\theta,\sigma^{2})$ and $\rho\sim\mathcal{N}(0,a^{2})$. Consider the following variable \begin{eqnarray*} L & = & b((\frac{1}{n}\sum_{i=1}^{i=n}r_{i})+\rho)^{2}, \end{eqnarray*} where $b,n$ are positive scalars. What are the characteristics of $L$ please? Does the probability density function of $L$ is $F$ non central please? Thanks.

Answer 1

Since $r_i \sim \text{IID } \mathcal{N} (\theta, \sigma^2)$ you have:

$$\bar{r}_n \equiv \frac{1}{n} \sum_{i=1}^n r_i \sim \mathcal{N} \Big( \theta, \frac{\sigma^2}{n} \Big).$$

Since $\rho \sim \mathcal{N} (0, a^2)$ you then have $\bar{r}_n + \rho \sim \mathcal{N} ( \theta, a^2 + \sigma^2 / n )$ so that:

$$\frac{\bar{r}_n + \rho}{\sqrt{a^2 + \sigma^2 / n}} \sim \mathcal{N} \Bigg( \frac{\theta}{\sqrt{a^2 + \sigma^2 / n}}, 1 \Bigg).$$

Hence, your random variable has a scaled non-central chi-squared distribution:

$$L \equiv b (\bar{r}_n + \rho)^2 \sim b (a^2 + \sigma^2 / n) \cdot \chi_1^2 \Big( \frac{\theta^2}{a^2 + \sigma^2 / n} \Big) .$$

Answer 2

1. $\frac{1}{n} \sum_{i=1}^n r_i \sim N(\theta, \sigma^2/n)$ (why?)
2. $\frac{1}{n} \sum_{i=1}^n r_i + \rho \sim N(\theta, a^2 + \sigma^2/n)$ (assuming $\rho$ is independent of the $r_i$)
3. $\sqrt{b} \left(\frac{1}{n} \sum_{i=1}^n r_i + \rho\right) \sim N(\sqrt{b}\theta, b(a^2 + \sigma^2/n))$
4. Square the above and note that it is a particular non-central chi squared random variable with 1 degree of freedom.