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100AHW4S

# 100AHW4S - STAT 100A HWIV Solution Problem 1 For Z...

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STAT 100A HWIV Solution Problem 1: For Z Bernoulli( p ), calculate E[ Z ]. A: E[ Z ] = 0 × (1 - p ) + 1 × p = p . Problem 2: For X Binomial( n,p ), calculate E[ X ]. A: We can represent X = Z 1 + Z 2 + ... + Z n , where Z i Bernoulli( p ) independently. E[ X ] = E[ Z 1 ] + ... + E[ Z n ] = np . Problem 3: For X Geometric( p ), calculate E[ X ]. A: Let T be the number of ﬂips until we get the ﬁrst head, then T Geometric( p ), P ( T = k ) = (1 - p ) k - 1 p , where k = 1 , 2 ,... Let q = 1 - p , then E[ T ] = X k =1 kP ( X = k ) = X k =1 kq k - 1 p = p X k =1 d dq q k = p d dq X k =1 q k = p d dq ( 1 1 - q - 1) = p 1 (1 - q ) 2 = 1 p . 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.

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100AHW4S - STAT 100A HWIV Solution Problem 1 For Z...

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