Prove that convergence almost everywhere implies convergence
Answer to relevant QuestionsConsider the random sequence Xn = X / (1 + n2), where is a Cauchy random variable with PDF, Determine which forms of convergence apply to this random sequence. Suppose, X1, X2, Xn, is a sequence of IID positive random variables. Define Show that as n →∞, yn converges in distribution, and find the distribution to which it converges. Consider the lottery described. (a) Assuming six million tickets are sold and that each player selects his/ her number independent of all others, find the exact probability of fewer than 3 players winning the lottery. (b) ...You collect a sample size N1 of data and find that a 90% confidence level has width, w. What should the sample size N2 be to increase the confidence level to 99.9% and yet maintain the same interval width, w? Suppose is a vector of IID random variables where each element has some PDF, fX (x). Find an example PDF such that the median is a better estimate of the mean than the sample mean.
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