100AHW4me

# 100AHW4me - X(3 What is the standard deviation of X How do...

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STAT 100A HWIV Due next Wed Problem 1: Suppose we roll a biased die, the probability mass function is p (1) = . 1, p (2) = . 1, p (3) = . 1, p (4) = . 2, p (5) = . 2, and p (6) = . 3. Let X be the random number we get by rolling this die. (1) Calculate Var( X ). (2) Suppose the rewards (in the amount of dollars) for the six numbers are respectively h (1) = - 20, h (2) = - 10, h (3) = 0, h (4) = 10, h (5) = 20, and h (6) = 100. Please calculate Var[ h ( X )]. Problem 2: Suppose in the population of voters, the proportion of those who would vote for a certain candidate is 20%. If we randomly sample 100 people from the population of voters. Let X be the number of people among these 100 people who would vote for this candidate. (1) What is the distribution of X ? (2) What is E( X ) and Var(
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Unformatted text preview: X )? (3) What is the standard deviation of X ? How do you interpret this standard deviation? Problem 3: For X ∼ Geometric( p ), calculate E( X ). Problem 4: 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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## This note was uploaded on 01/27/2011 for the course STATISTICS 100a taught by Professor Cristou during the Winter '10 term at UCLA.

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