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2010100AHW5-1

# 2010100AHW5-1 - T> 5 Problem 3 Suppose we divide the time...

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STAT 100A HWV Problem 1: Suppose in the population of voters, the proportion of those who would vote for a certain candidate is 20%. (1) If we randomly sample 5 people from the population of voters, what is the probability that we get at least 2 supporters of this candidate? (2) If we randomly sample 100 people from the population of voters. Let X be the number of people among these 100 people who support this candidate. What is the probability that X > 28? You only need to write down this probability. You do not need to calculate it. (3) What is E( X ) and Var( X )? What is the standard deviation of X ? Problem 2: For T Geometric( p ), calculate E( T ). If p = . 2, then what is the probability that
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Unformatted text preview: T > 5? Problem 3: 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 . (3) Please give a real example of X and T with concrete value of λ and its unit. What is the meaning of λ ? 1...
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