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EECE7204

Problem Set 6

Problem 1 (4.32). Let X ∼ N (0, σ2). Show that

Problem 2 (4.36). Let E{Xi} = µ, Var{Xi} = σ 2 . We wish to estimate µ with the sample mean

Compute the mean and variance of ˆµN ; assuming the Xi for i = 1, . . . , N are independent.

Problem 3 (4.37). In the previous problem, how large should N be so that

Problem 4 (4.45). Let X have a Cauchy pdf

Compute the CF ΦX(ω) of X.

Problem 5 (4.48). Let

where the Xi are i.i.d. Cauchy RVs with

Show that the pdf of Y is

that is, is identical to the pdf of the Xi ’s and independent of N. (Hint: Use the CF approach.)

Problem 6. (Confidence interval) Matlab file hwk-confint.mat contains independent observations of random variable X. Your task is to estimate the pdf of X. To do so, divide the range of observed values into 25 bins of equal length, and count the number of values that fall in each bin. Plot the estimated pdf along with 90% confidence intervals.