Variance of a sum of Brownian motion

Let WtW_t be a standard Brownian motion. Compute Var⁡(W1+W2)\operatorname{Var}(W_1 + W_2).

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  1. W1W_1 and W2W_2 are dependent. Use Var⁡(A+B)=Var⁡A+Var⁡B+2Cov⁡(A,B)\operatorname{Var}(A+B)=\operatorname{Var}A+\operatorname{Var}B+2\operatorname{Cov}(A,B).
  2. For Brownian motion Cov⁡(Ws,Wt)=min⁡(s,t)\operatorname{Cov}(W_s,W_t)=\min(s,t) - here that covariance is 11.

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