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Unformatted text preview: MA 316: MATHEMATICAL PROBABILITY ASSIGNMENT #12 SOLUTIONS: SECTIONS 5.4 AND 6.1 FALL 2003 5.20: If ¯ X is the mean of a random sample of size 100 from a distribution that is χ 2 (50) then α = r/ 2 = 25 and β = 2 so that μ = αβ = 50 and σ 2 = αβ 2 = 100 so that P (49 < ¯ X < 51) = P ˆ (49 50) √ 100 10 < Z < (51 50) √ 100 10 ! = P ( 1 < Z < 1) = 2Φ(1) 1 = 2(0 . 841) 1 = 0 . 682 5.23: If X 1 ,X 2 ,...,X 15 is a random sample of size 15 from a distribution having density function f ( x ) = 3 x 2 , < x < 1 then for each X i we know that μ X i = E [ X i ] = Z 1 x (3 x 2 ) dx = 3 / 4 and E [ X 2 i ] = Z 1 x 2 (3 x 2 ) dx = 3 / 5 so that σ 2 X i = 3 / 5 (3 / 4) 2 = 3 / 80 . If ¯ X is the mean of this random sample then for ¯ X we have μ = 3 / 4 and σ = σ X i / √ n = r 3 80 / √ 15 so that P ( 3 / 4 < ¯ X < 4 / 5 ) = P 3 / 5 3 / 4 q 3 80 / √ 15 < ¯ X 3 / 4 q 3 80 / √ 15 < 4 / 5 3 / 4 q 3 80 / √ 15 = P ( 3 < Z < 1) = Φ(1) (1 Φ(3)) = 0 . 840 . 5.24: If Y denotes the sum of observations of a random sample X 1 ,X 2 ,...X 12 of size 12 from a distribution having density f ( x ) = 1 / 6 for x = 1 , 2 , 3 , 4...
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This note was uploaded on 09/10/2009 for the course STATS 517 taught by Professor Song during the Fall '07 term at Purdue.
 Fall '07
 Song
 Probability

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