Given data, $p(y_1,...,y_n|\theta)$ normally distributed with mean, $\mu=1$ and unknown variance, $\sigma^2$ and the prior$$\theta \sim Beta \ (1,1) \implies p(\theta)=1$$ Find the posterior distribution, $p(\theta|y_1,...,y_n)$.
I tried and got the posterior $$p(\theta|y_1,...,y_n)\sim N(1,\frac{\sigma^2}{n})$$
Don't know if it's correct.