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This ratio is called the likelihood ratio test statistic. The test procedure calls for rejecting the null hypothesis H0 when the value of this ratio is small, say, whenever k, where k is a constant. Thus, the likelihood ratio principle requires rejecting H0 when L( 1) is much larger than L( 0), which would indicate that the sample data are more compatible with the alternative hypothesis H1 than with the null hypothesis H0. Usually, the constant k would be selected to give a speci ed value for , the type I error probability. These ideas can be illustrated by a hypothesis-testing problem that we have studied before that of testing whether the mean of a normal population has a speci ed value 0. This is the one-sample t-test of Section 9-3. Suppose that we have a sample of n observations from a normal population with unknown mean and unknown variance 2, say, X1, X2, p , Xn. We wish to test the hypothesis H0: 0 versus H1: 0. The likelihood function of the sample is a

Furthermore, suppose the messages in Table 25 form a 3-legged depth Then the permutation at each position of the three messages is the same Let Pi be the ith permutation generated by this cipher for this particular key Then we have some information on the first several permutations For example, we know that Po maps plaintext P to ciphertext J , plaintext T to ciphertext K and plaintext M to ciphertext X Also, PI maps E to K and 0 to P, and I to H, and so on In this way, we can partially reconstruct the permutations, and the more legs of depth that are available, the more information on each permutation we obtain While it was clear to Rowlett and his team that Purple was a substitution cipher, it was unclear how the permutations were generated.

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You can change different options for different shooting scenarios in the Shooting menu. As you begin to get more accustomed to your PowerShot G11, you will probably enter this menu fairly often to change settings or try different ones. I encourage you to explore all of the creative possibilities your camera has to offer.

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{ : }, respectively. The values 0 and 1 are the usual maximum likelihood estimates for and 2:

The cryptanalysts had knowledge of previous Japanese cipher machines, as well as knowledge of other cryptographic devices of the time, including rotor machines The knowledge of an earlier Japanese cipher known as Red proved most valuable The Red cipher employed an unusual split of the alphabet into sixes and twenties, which was carried over into Purple Initially, the Red cipher split the alphabet into the six vowels, AEIOUY, and the remaining twenty consonant Substituting vowels for vowels and consonants for consonants was also used in some other ciphers of the time This split reduced the cost of cabling the ciphertext messages, since the resulting messages were considered pronounceable ---even though the ciphertext was gibberish--and therefore were charged a lower rate than messages consisting of random letters [52].

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But encrypting vowels to vowels is a serious weakness, since some niessages can be inferred simply based on the placement of vowels within the ciphertext The Japa,nese apparently realized this was a weakness and usage of the Red cipher was modified so that any six letters could act as the sixes This is precisely the same situation as with Purple The ciphertext sixes would be expected to each occur with the average probability of the corresponding plaintext sixes, and similarly, each ciphertext twenties let,ter would occur with t,he average probability of the plaintext.

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