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Functions of Random Variables Often we have to consider random variables which are functions of other random variables. Let X be a random variable and g (.) is a function. Then Y g ( X ) is a random variable. We are interested to find the pdf of Y . For example, suppose X represents the random voltage input to a full-wave rectifier. Then the rectifier output Y is given by Y X . We have to find the probability description of the random variable Y . We consider the following cases: (a) X is a discrete random variable with probability mass function p X ( x) The probability mass function of Y is given by pY ( y ) P(Y y) P( x | g ( x) y) P( X x) p X ( x) x| g ( x ) y x| g ( x ) y (b) X is a continuous random variable with probability density function y g ( x) is one-to-one and monotonically increasing The probability distribution function of Y is given by f X ( x) and FY ( y ) P Y y P g ( X ) y P X g 1 ( y ) P( X x) x g 1 ( y ) FX ( x) x g 1 ( y ) fY ( y ) dFY ( y ) dy dFX ( x) dy x g 1 ( y ) dFX ( x) dx dx dy x g 1 ( y ) dFX ( x) dx dy dx x g 1 ( y ) fY ( y ) f X ( x) g ( x) x g 1 ( y ) f X ( x) f X ( x) dy g ( x) x g 1 ( y ) dx This is illustrated in Fig. Example 1: Probability density function of a linear function of a random variable Suppose Y aX b, a 0. y b dy Then x and a a dx y b fX ( ) f X ( x) a fY ( y ) dy a dx Example 2: Probability density function of the distribution function of a random variable Suppose the distribution function FX ( x) of a continuous random variable X is monotonically increasing and one-to-one and define the random variable Y FX ( X ). Then, fY ( y) 1 0 y 1. y FX ( x) Clearly 0 y 1 dy dFX ( x) f X ( x) dx dx f ( x) f X ( x) fY ( y ) X 1 dy f X ( x) dx fY ( y ) 1 0 y 1. Remark (1) The distribution given by fY ( y) 1 0 y 1 is called a uniform distribution over the interval [0,1]. (2) The above result is particularly important in simulating a random variable with a particular distribution function. We assumed FX ( x) to be one-to-one function for invariability. However, the result is more general- the random variable defined by the distribution function of any random variable is uniformly distributed over [0,1]. For example, if X is a discrete RV, FY ( y ) =P(Y y ) P( FX ( x) y ) P( X FX1 ( y )) FX ( FX1 ( y )) y ( Assigning FX1 ( y ) to the left-most point of the interval for which FX ( x) y ). dF ( y ) fY ( y ) Y 1 0 y 1. dy Y FX ( X ) y x FX1 ( y ) X (c) X is a continuous random variable with probability density function y g ( x) has multiple solutions for x Suppose for y Y , y g ( x) has solutions xi , i 1, 2,3,............., n . Then f X ( x) and n fY ( y ) i 1 f X ( x) dy dx x xi Proof: Consider the plot of Y g ( X ) . Suppose at a point y g ( x) , we have three distinct roots as shown. Consider the event y Y y dy . This event will be equivalent to union events x1 X x1 dx1 ,x2 dx2 X x2 and x3 X x3 dx3 P y Y y dy P x1 X x1 dx1 P x2 dx2 X x2 P x3 X x3 dx3 fY ( y)dy f X ( x1 )dx1 f X ( x2 )(dx2 ) f X ( x3 )dx3 Where the negative sign in dx2 is used to account for positive probability. Therefore, dividing by dy and taking the limit, we get dx dx dx fY ( y ) f X ( x1 ) 1 f X ( x2 ) 2 f X ( x3 ) 3 dy dy dy f X ( x1 ) 3 i 1 dx dx1 dx f X ( x2 ) 2 f X ( x3 ) 3 dy dy dy f X ( xi ) dy dx x xi In the above, we assumed y g ( x) to have three roots. In general, if y g ( x) has n roots, then n fY ( y ) i 1 f X ( xi ) dy dx x xi Example 3: Probability density function of a linear function of a random variable Suppose Y aX b, a 0. y b dy Then x and a a dx y b fX ( ) f X ( x) a fY ( y ) dy a dx Example 4: Probability density function of the output of a full-wave rectifier Suppose Y X , a X a, a0 Y y y y y x has two solutions x1 y and x2 y and fY ( y ) f X ( x ) x y X dy 1 at each solution point. dx f X ( x)x y 1 1 f X ( y ) f X ( y ) Example 5: Probability density function of the output of a square-law device Y cX 2 , c 0 y cx 2 And x y c y0 dy dy 2cx so that 2c y / c 2 cy dx dx fY ( y ) fX y / c fX 2 cy = 0 otherwise y /c y0