structural:summarization
Basic properties

The unconditional probability density function follows a normal distribution with mean = 0 and variance = t, at a fixed time t: f W t (x) = 1 2 π t e − x 2 / (2 t). {\displaystyle f_{W_{t}}(x)={\frac {1}{\sqrt {2\pi t}}}e^{-x^{2}/(2t)}.}

The expectation is zero: E = 0. {\displaystyle \operatorname {E} =0.}

The variance, using the computational formula, is t: Var (W t) = t. {\displaystyle \operatorname {Var} (W_{t})=t.}

These results follow immediately from the definition that increments have a normal distribution, centered at zero. Thus W t = W t − W 0 ∼ N (0, t). {\displaystyle W_{t}=W_{t}-W_{0}\sim N(0,t).}