what are the characteristics of this random variable? 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. 
 A: *

*$\frac{1}{n} \sum_{i=1}^n r_i \sim N(\theta, \sigma^2/n)$ (why?)

*$\frac{1}{n} \sum_{i=1}^n r_i + \rho \sim N(\theta, a^2 + \sigma^2/n)$ (assuming $\rho$ is independent of the $r_i$)

*$\sqrt{b} \left(\frac{1}{n} \sum_{i=1}^n r_i + \rho\right) \sim N(\sqrt{b}\theta, b(a^2 + \sigma^2/n))$

*Square the above and note that it is a particular non-central chi squared random variable with 1 degree of freedom.

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