Question: The objective of this problem is to know the construction of Brownian motion from a symmetric random walk. Problem: Consider the random variables X,

The objective of this problem is to know the construction of Brownian

The objective of this problem is to know the construction of Brownian motion from a symmetric random walk. Problem: Consider the random variables X, as follows. X = 8 if the kth coin toss results in heads, and X = - if the kth coin toss results in tails. Thus, 8 with probability 2 X = with probability Where 8 is the length of each subinterval corresponds to a time slot of length 8. -8 6 26 36 46 56 66 76 86 96 106 He wad 1. Determine E(X) and Var(X). 2. Define the process W(1) as follow. Let W(0)-0 and at time - 8n, the value of W() is given by W(t)=W (8n) a) Calculate E(W(t)) and Var(W(t)) for any time t (0,00). b) Apply Central Limit Theorem to show that W(t)-N(0,t) with continious smples paths X (PI)

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