1. Consider a probability mass function Px(2) for an M-ary alphabet Ax = {21, ..., IM} with pi 4 P(X = x;), i = 1,..., M

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1 Consider A Probability Mass Function Px 2 For An M Ary Alphabet Ax 21 Im With Pi 4 P X X I 1 M 1 (41.21 KiB) Viewed 39 times

1. Consider a probability mass function Px(2) for an M-ary alphabet Ax = {21, ..., IM} with pi 4 P(X = x;), i = 1,..., M. Now suppose that we have available N i.i.d drawings X1,..., Xy from Px(x) (with very large N) using which we wish to estimate the unknown probability mass function. The most straightforward way to do this is to compute the fraction of each realization. In particular, let 1. (X) be an indicator function defined by 1. (X) = { 1 if X = xi, 0 otherwise. Then the estimate of p, is given by 1-1 1:(X) Pi= N Using the weak law of large numbers, show that pi converges to pi in probability. * = E(X)