Get probability density function using mean and variance parameters?

Question

I derived the mean and variance parameters for a normal distribution. How would I write the probability density function using this information?

y|x1, x2, and (theta) ~ N(A, sigma^2)

  A = E[y] = (theta0) + (theta1)(x1) + (theta2)(x2) + (theta3)(x1)^2 

  var[y] = sigma^2

Answer 1

The density of a normal distribution with mean $A$ and variance $\sigma^2$ is $$f(y \mid A, \sigma^2)=\frac{1}{\sqrt{2 \pi \sigma^2}} e^{-\frac{(y-A)^2}{2 \sigma^2}}$$ which you could write as $$f(y \mid \theta_0,\theta_1,\theta_2,\theta_3,x_1,x_2, \sigma^2)=\frac{1}{\sqrt{2 \pi \sigma^2}} e^{-\frac{\left(y-\theta_0-\theta_1 x_1-\theta_2 x_2 -\theta_3 x_1^2\right)^2}{2 \sigma^2}}$$