(a) Let X, X,..., X, be independent N(0, 1) random variables and X, be their sample...

  

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(a) Let X₁, X₂,..., X₁, be independent N(0, 1) random variables and X, be their sample mean. For n = 10, 100, and 1000, simulate this problem 2000 times and plot the histogram of the values of X, (you need to plot three histograms; one for each choice of n). How does Var (X) relate to the histogram of the values of X₁? 2 (b) In a symmetric random walk on a line, at step n = 0, we start from point Xo = 0. At each step n, we flip an unbiased coin and set Xn+1 X₂ + 1 Heads Xn-1 Tails (3) By running the random walk 1000 times, for n = {1,..., 50), estimate Var (X₁) and plot it as a function of n. Do the same for E(Xn). (c) For the previous part, compute E(X) and Var (X₂) theoretically and compare them with your plots. (a) Let X₁, X₂,..., X₁, be independent N(0, 1) random variables and X, be their sample mean. For n = 10, 100, and 1000, simulate this problem 2000 times and plot the histogram of the values of X, (you need to plot three histograms; one for each choice of n). How does Var (X) relate to the histogram of the values of X₁? 2 (b) In a symmetric random walk on a line, at step n = 0, we start from point Xo = 0. At each step n, we flip an unbiased coin and set Xn+1 X₂ + 1 Heads Xn-1 Tails (3) By running the random walk 1000 times, for n = {1,..., 50), estimate Var (X₁) and plot it as a function of n. Do the same for E(Xn). (c) For the previous part, compute E(X) and Var (X₂) theoretically and compare them with your plots.

Answer rating: 100% (QA)

A To solve the sample mean X n for n 10 100 and 1000 we can use the following Python code The histograms show that the distribution of X n becomes increasingly concentrated around 0 as n increases Thi... View the full answer