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Let X(t) be a Brownian motion process with drift = 10 and volatility = 2.

(i) Find the conditional mean of X(2) given X(1) = 0 and X(4) = 2.

(ii) Find the conditional variance of X(2) given X(1) = 0 and X(4) = 2.

(iii) Find the conditional distribution of X(2) given X(1) = 0 and X(4) = 2.

(iv) For s < t, derive the conditional variance of X(s) given X(t) and X(0) as a function of (s, t).

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