linear regression, expectation and mean squared error

Let us assume that data is generated according to a true model $$y_i = \beta_{true}x_i + \epsilon_i$$for $i = 1, ..., n$

Assume that $x_i$ are fixed, and $\epsilon_i$~ N(0, $\sigma^2$) independently.

Let $$\hat\beta =\frac{\sum^{n}_{i=1}y_ix_i}{\sum^{n}_{i=1}x^2_i + \lambda}$$ be the shrinkage estimator from the ridge regression.

How to calculate expectation and variance of $\hat\beta$, and mean squared error E$[(\hat\beta - \beta_{true})^2]$ ?

I'm stuck on this part for expectation. What to do next?$$E(\hat\beta)= E(\frac{\beta_{true}\sum^{n}_{i=1}x_i^2 + \sum^{n}_{i=1}x_i\epsilon_i}{\sum^{n}_{i=1}x^2_i + \lambda}) = (E(\frac{\beta_{true}\sum^{n}_{i=1}x_i^2 }{\sum^{n}_{i=1}x^2_i + \lambda})$$