Quiz1 - 36-226 Summer 2010 Quiz 1 July 2 1. Indicate...

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Unformatted text preview: 36-226 Summer 2010 Quiz 1 July 2 1. Indicate whether the following are TRUE or FALSE. Write the entire word below each statement. Do not assume ANYTHING not written in the problem statement. (a) All estimators are statistics. TRUE ˆ ˆ (b) If θ is a method of moments estimator (MOME) for θ, then E[θ] = θ. FALSE (c) The mean is effected less by outliers than the median. FALSE (d) If X1 , . . . , Xn are identically distributed, each with density fX (x), then the joint density of X1 , . . . , Xn is given by n fX1 ,...,Xn (x1 , . . . , xn ) = fX (x). i=1 FALSE (e) Let X1 , . . . , Xn be a random sample from a density fX (x). Then, 1 E n n ¯ (Xi − X )2 = E[(X − E[X ])2 ]. i=1 FALSE 2. Let X1 , . . . , Xn be iid Geometric(p), with fX (x) = p(1 − p)x−1 x = 1, 2, 3, . . . 0 else Find an MOME for P(X ≤ 2). Useful result: E[X ] = 1/p and V ar[X ] = 1−p . p2 There are many MOMEs one could find. Here is one: First we find the MOME for p, p. ˆ ¯ X = E (X ) = 1/p ˆ ¯ ⇒ p = 1/X ˆ Now, P(X ≤ 2) = P(X = 1) + P(X = 2) = p(1 − p)1−1 + p(1 − p)2−1 = p + p(1 − p) = p[1 + (1 − p)] = p(2 − p). ˆ Using the result that the MOME for g (θ) is g (θ) means that the MOME for P(X ≤ 2) is given by 1 1 P(X ≤ 2) = p(2 − p) = ¯ 2 − ¯ . ˆ ˆ X X 1 ...
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This document was uploaded on 07/14/2011.

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