I have a question concerning the distribution of the first hitting time of Brownian Motion $\tau_x = \inf_{t\geq 0}\{W_t=x\}$, where $W_t$ is Brownian motion. Using some calculus, I found out that the density of $\tau_x$ should be $$f(t)=\frac{x}{\sqrt{2\pi}t^{\frac{3}{2}}}\exp(-\frac{x^2}{2t})$$
In some scripts in the internet I have seen that this seems to be right. However, the task was to proove that $\tau_x$ has an inverse gaussian distribution, e.g. the density should be of the form
$$f(x)=(\frac{\lambda}{2\pi x^3})^{\frac{1}{2}}\exp(-\frac{\lambda(x-\mu)^2}{2\mu^2 x})$$ and I think it's impossible to get the upper density into that form since I miss a $t$ in the numerator of my exponent. Can anyone give me a reason?