EECE7204 Problem Set 6
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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.
2023-12-05