PROBABILITY AND STATISTICS FOR ENGINEERING Hossein Sameti Department of Computer Engineering Sharif University of Technology Functions of a Random Variable

: X : a r.v defined on the model g ( X ): a function of the variable x,  Is Y necessarily a r.v?  If so what is its PDF pdf 2

Functions of a Random Variable 3

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 For and hence Example:  For And We conclude: 5

 For according to the figure,  the event is equivalent to Example:  For 6

Example: - continued  If represents an even function, 7

Example: - continued  If so that  We obtain the p.d.f of to be  which represents a Chi-square r.v with n = 1, since  Thus, if X is a Gaussian r.v with then represents a Chi-square r.v with one degree of freedom ( n = 1). 8

Example  For:  We have  For we have and so that 9

Example – continued  Thus, 10

Example: Half-wave Rectifier  Consider  In this case  For since  Thus, 11

Continuous Functions of a Random Variable  A continuous function g ( x )  nonzero at all but a finite number of points  has only a finite number of maxima and minima  eventually becomes monotonic as  Consider a specific y on the y -axis, and a positive increment 12

Continuous Functions of a Random Variable  For where is of continuous type,  has three solutions when  X could be in any one of three disjoint intervals: So, 13

Continuous Functions of a Random Variable  For small we get  Here, and so  As  If the solutions are all in terms of y, the right side is only a function of y. 14

Example Revisited  For all and  for.  Moreover  and using, which agrees with previous solution. 15

Example:  Find  Here for every y, is the only solution, and  and substituting this into, we obtain 16

Example  Suppose and Determine  X has zero probability of falling outside the interval  has zero probability of falling outside  outside this interval.  For any the equation has an infinite number of solutions where is the principal solution.  using the symmetry we also get etc. Further,  so that 17

Example – continued  for  In this case  (Except for and the rest are all zeros). (a) (b) 18

Example – continued  Thus, 19

Example:  As x moves in y moves in  The function is one-to-one for  For any y, is the principal solution. (Cauchy density function with parameter equal to unity) 20

Functions of a Discrete-type R.V  Suppose X is a discrete-type r.v with  and  Clearly Y is also of discrete-type, and when and for those s, 21

Example  Suppose so that  Define Find the p.m.f of Y.  Solution: X takes the values  Y only takes the values  so that for 22