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For the second integral, due to dV being negative, the original integral would have been over [V + dV, V ] which is equivalent to the integral in the expression above after interchanging the limits of integration and multiplying the integral by 1 The above integral relationships are equivalent to pw (W )dW which can be rewritten as pv (V ) pv (V ) dW dV = pw (f (V )) |f (V )| = pw (f (V )) = pv (V )|dV |

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Issues related to the transformation of random variables are clari ed by the following examples Example 44 Find the density for y where y = ax + b, a = 0, and x N (0, 2 ) 1 In this case, f (y) = y b and f (y) = a Therefore, a py (Y ) = = 1 Y b px |a| a (Y b)2 1 exp 2(a )2 2 |a|

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Figure 43: Graph of y = |x| with variables de ned for Example 45 Note that Py (Y ) has the form of a Gaussian distribution with mean b and variance (a )2 ; therefore, y is a Gaussian random variable In general, all linear operations on Gaussian random variables result in Gaussian random variables Example 45 Find the density for y where y = |x| and x N (0, 2 ) Because the function g is not invertible, the result of eqn (416) cannot be used Instead, we have to revert to the main idea of eqn (415) which is that the change of variable must preserve the probability of events Consider the variables de ned in Figure 43 For this problem, we have P {Y0 y Y0 + dY } = P {X2 + dX2 x X2 } + P {X1 x X1 + dX1 } where dY represents a positive di erential change from Y0 and X1 = Y0 X2 = Y0 X1 + dX1 = Y0 + dY X2 + dX2 = Y0 dY

The name of the album from which the track comes The album artist assigned for the track An integer giving the album s overall rating, from 0 (no stars) to 100 (five stars) The artist assigned for the track An integer giving the track s bitrate in Kbps This property is read-only A real number giving the position at which the track is bookmarked for resuming play true if the track s playback position will be remembered for resuming play, false if it will not be An integer giving the track s tempo in beats per minute The category assigned to the track Any comments you ve added to the track s Comments field true to have iTunes handle this track as being part of a compilation, false to handle it as being part of a normal album (supposing such a thing exists) The composer credited with the track The playlist that contains the track for example, user playlist id 28947 of source id 49 of application iTunes An integer giving the track s unique ID within iTunes This property is read-only The date and time the track was added to iTunes This property is readonly The text from the Description field (on the Video tab of the Item Information dialog box) An integer giving the number of discs in the album An integer giving the number of this track s disc within the album (for example, 1 for disc 1 of 2) A real number giving the track s duration in seconds for example, 146703002929688 This property is read-only true if the track s check box is selected for playback, false if the check box is cleared The episode ID for the track (for video shows) The episode number of the track (for video shows) The name of the equalizer preset to use when playing the track A real number giving the stop time of the track in seconds for example, 287476989746094.

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composer container database ID date added description disc count disc number duration enabled episode ID episode number EQ finish

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