Suppose $X_i$ ($i=1,2,3,\ldots, n$) are i.i.d random variables, each having an inverse Gaussian distribution with parameters $\lambda$ and $\mu$. I have obtained the following for the MLEs of $\lambda$ and $\mu$: $$\hat{\mu} = \bar{X}=\frac{1}{n}\sum_{i=1}^{n}X_i$$ $$\hat{\lambda} = \frac{n}{V}, V = \sum_{i=1}^n \left(\frac{1}{X_i} - \frac{1}{\hat{\mu}}\right)$$
I was interested to see if these estimates are biased or unbiased. In the case of $\hat{\mu}$, we have $$E(\hat{\mu})=\frac{1}{n}\sum_{i=1}^n E(X_i)=\mu$$ so clearly this is unbiased for $\mu$. Here is my question:
How do I determine whether $\hat{\lambda}$ is a biased estimator for $\lambda$?
In particular, what is the distribution of the estimator? According to Wikipedia, it's chi-squared, but how do I prove it?