ma316f03hw8

ma316f03hw8 - . 4.5: If X has density function f ( x ) = 4...

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MA 316: MATHEMATICAL PROBABILITY ASSIGNMENT #8 SOLUTIONS: SECTIONS 3.3, 4.1 FALL 2003 3.35: If X is χ 2 (5) and P ( c < X < d ) = 0 . 95 and P ( X < c ) = 0 . 025 then P ( X < d ) - P ( X < c ) = P ( X < d ) - 0 . 25 = 0 . 95 so P ( X < d ) = 0 . 975. From the tables in the back of the text, this implies that d = 12 . 8 and similarly that c = 0 . 831 . 3.36: If X is gamma with α = 3 and β = 4 then Y = X/ 2 is χ 2 (2 α ) = χ 2 (6) so P (1 . 64 < Y < 12 . 6) = P (1 . 64 < X/ 2 < 12 . 6) = P ( X/ 2 < 12 . 6) - P ( X/ 2 < 1 . 64) = 0 . 95 - 0 . 05 = 0 . 90 from the table in the back of the text. 4.2: For an individual random variable X from the random sample, we know that P ( X > 0) = Z 1 0 ( x + 1) / 2 dx = 3 / 4 so by independence the probability that exactly four observations of a random sample of size 5 from this distribution exceed zero is ± 5 4 ¶± 3 4 4 ± 1 4 = 405 1024 . 4.3: If X 1 ,X 2 ,X 3 is a random sample of size 3 from a distribution that is N (6 , 4) then the event that the largest is greater than 8 is the complement of the event that all three are less than 8. This gives by independence 1 - P ( X 1 < 8 ,X 2 < 8 ,X 3 < 8) = 1 - P ( X 1 < 8) P ( X 2 < 8) P ( X 3 < 8) = 1 - Φ ± 8 - 6 2 ¶‚ 3 = 1 - (Φ(1)) 3 = 1 - (0 . 841) 3 = 0 . 40518
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Unformatted text preview: . 4.5: If X has density function f ( x ) = 4 x 3 for 0 &lt; x &lt; 1 and Y =-2ln X 4 then the distribution function for Y is F ( y ) = P ( Y y ) = P (-2ln X 4 y ) = P (4ln X =-y/ 2) = P ( X &gt; e-y/ 8 ) which reduces to Z 1 e (-y/ 8) 4 x 3 dx = 1-e-y/ 2 so f ( y ) = F ( y ) = e-y/ 2 2 , which is the density function for the random variable 2 (2). 4.11: If X 1 and X 2 are random variables from the distribution N (0 , 1) and Y = X 2 1 + X 2 2 then F ( y ) = P ( Y y ) = P ( X 2 1 + X 2 2 y ) Z Z x 2 1 + x 2 2 y 1 2 e-x 2 1 2-x 2 2 2 dx 1 dx 2 . 1 Changing to polar coordinates reduces this integral to 1 2 * Z 2 Z y e-r 2 / 2 rdrd = 1-e-y/ 2 so that f ( y ) = F ( y ) = 1 2 e-y/ 2 which is the density function for a random variable with 2 (2) distribution. 2...
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ma316f03hw8 - . 4.5: If X has density function f ( x ) = 4...

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