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This joint ensemble has the special property that its two marginal distributions, P (x) and P (y), are identical. 1. 3). The probability P (y | x = q) is the probability distribution of the second letter given that the first letter is a q. 3. Conditional probability distributions. (a) P (y | x): Each row shows the conditional distribution of the second letter, y, given the first letter, x, in a bigram xy. (b) P (x | y): Each column shows the conditional distribution of the first letter, x, given the second letter, y.

24). No, they are not independent. f. 3). 27). We define the fraction fB ≡ B/K. (a) The number of black balls has a binomial distribution. P (nB | fB , N ) = N f nB (1 − fB )N −nB . 58) var[nB ] = N fB (1 − fB ). 1). The standard deviation of nB is var[nB ] = N fB (1 − fB ). When B/K = 1/5 and N = 5, the expectation and variance of nB are 1 and 4/5. 89. When B/K = 1/5 and N = 400, the expectation and variance of nB are 80 and 64. The standard deviation is 8. 27). 59), which is by definition the expectation of the numerator.

0 1 0 0 1 1 1 1 0 1 0 0 1 1 1 1 0 1 0 0 1 This matrix is ‘redundant’ in the sense that the space spanned by its rows is only three-dimensional, not seven. This matrix is also a cyclic matrix. Every row is a cyclic permutation of the top row. Cyclic codes: if there is an ordering of the bits t1 . . tN such that a linear code has a cyclic parity-check matrix, then the code is called a cyclic code. The codewords of such a code also have cyclic properties: any cyclic permutation of a codeword is a codeword.