Any coefficient, $\beta_0 \in [\beta_L, \beta_U]$ is modeled in the Beta distribution and its pdf is given by
$$p(\beta_0) = \frac{\Gamma(\lambda_1+\lambda_2)}{(\beta_U-\beta_L)\Gamma(\lambda_1)\Gamma(\lambda_2)}(\frac{\beta_0-\beta_L}{\beta_U-\beta_L})^{\lambda_1-1}(1-\frac{\beta_0-\beta_L}{\beta_U-\beta_L})^{\lambda_2-1},$$
where $\lambda_1$, $\lambda_2$ are shape parameters and symbol $\Gamma$ represents the Gamma function.
The following parameters are used: $\lambda_1 = \lambda_2 = 1.1$, the lower limit $\beta_L = 10,000$, and the upper limit $\beta_U = 63,000$. Probability density of $\beta_0$ is shown below. 
$\beta_0$">
Our main concern here is to approximate $p(\beta_0)$ by a Gaussian density. Any suggestion will be great help to me.