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100AHW4 - Problem 5 Suppose we divide the time axis into...

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STAT 100A HWIV Due next Wed in class Problem 1: For Z Bernoulli( p ), calculate E[ Z ]. Problem 2: For X Binomial( n,p ), calculate E[ X ]. Problem 3: For X Geometric( p ), calculate E[ X ]. Problem 4: Suppose we have a ﬁve-letter alphabet, A, B, C, D, E, and their probabilities are p ( A ) = 1 / 4, p ( B ) = 1 / 4, p ( C ) = 1 / 4, p ( D ) = 1 / 8, p ( E ) = 1 / 8. (1) Design a scheme for generating a letter according to the above distribution by coin ﬂipping. (2) Calculate the expected number of coin ﬂippings for generating a letter according to the above distribution. (3) Argue that your coin ﬂipping scheme leads to a preﬁx coding of the ﬁve letters.
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Unformatted text preview: Problem 5: Suppose we divide the time axis into small periods (0 , Δ t ), (Δ t, 2Δ t ), . .. Within each period, we ﬂip a coin independently. The probability of getting a head is λ Δ t . (1) Let X be the number of heads within the interval [0 ,t ]. Calculate the limit of P ( X = k ) as Δ t → 0, for k = 0 , 1 , 2 ,... . Also calculate E[ X ]. (2) Let T be the time until the ﬁrst head. Calculate the limit of P ( T > t ) as Δ t → 0. In both (1) and (2), let us assume that t is a multiple of Δ t . 1...
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