Unformatted text preview: x dx. We now need to integrate by parts again, with f 3 ( x ) = 2 x and g 3 ( x ) = cos x so that f 3 ( x ) = 2 and G 3 ( x ) = sin( x ). This gives us Z 2 π 3 π/ 2 x 2 (sin x ) dx = x 2 cos x ± ± 2 π 3 π/ 22 x sin x  2 π 3 π/ 2Z 2 π 3 π/ 2 2 sin( x ) dx ! = x 2 cos x ± ± 2 π 3 π/ 2[2 x sin x + 2 cos( x )] 2 π 3 π/ 2 = ² x 2 cos x2 x sin x2 cos( x ) ³ 2 π 3 π/ 2 = ( (2 π ) 2 cos 2 π2(2 π ) sin 2 π2 cos(2 π ) )´ 3 π 2 µ 2 cos 3 π 22 · 3 π 2 sin 3 π 22 cos ´ 3 π 2 µ ! = ( 4 π 2 · 14 π ·2 · 1 )´ 9 π 2 4 ·2 · 3 π 2 · (1)2 · µ = 4 π 23 π2 Therefore, var( X ) = Z 2 π 3 π/ 2 x 2 f ( x ) dxE ( X ) 2 = (4 π 23 π2)(2 π1) 2 = (4 π 23 π2)(4 π 24 π + 1) = π3 . 1...
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 Fall '08
 Fretchet
 Probability theory, Cos, probability density function

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