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Unformatted text preview: HW 3 for Stat 7249  Spring 2006 Due February 20 Reading in text for this assignment • Chapter 4 Datasets • posted on class web page 1. Show that the first four cumulants of Z = Y mπ √ mπ (1 π ) , where Y ∼ Bin ( m, π ) are 0, 1, O ( m 1 / 2 ), and O ( m 1 ), respectively. This implies that for fixed π , as m → ∞ , the cumulants of Z approach those of a standard normal random variable (so converge in distribution to a standard normal). Also, derive the first four cumulants of a standard normal random variable to verify they are 0,1, 0, and 0, respectively. 2. Suppose that Y 1 and Y 2 are independent Poisson random variables with means μ and ρμ respectively. Derive the distribution of [ Y 1  Y 1 + Y 2 = m ]. 3. Variance stabilizing transformations: Suppose that Y ∼ Bin ( m, π ) with m large. By expand ing in a Taylor series, show that the random variable, Z = arcsin { ( Y/m ) 1 / 2 } has approximate first two moments E [ Z ] ≈ arcsin ( π 1 / 2 ) 1 2 π 8...
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This note was uploaded on 01/15/2012 for the course STA 7249 taught by Professor Daniels during the Spring '08 term at University of Florida.
 Spring '08
 Daniels

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