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Unformatted text preview: PE RANDOM VARIABLES FX 1 0 1 2 3 4 5 6 Figure 3.1: CDF for the roll of a fair die.
The notions of left limit and right limit are useful for discussing functions with jumps, such
as CDFs. Given a function F on the real line and a value x, the left limit of F at x, denoted by
F (x−), is the limit of F (y ) as y converges to x from the left. Similarly, the right limit of F at x,
denoted by F (x+), is the limit of F (y ) as y converges to x from the right. That is,
F (x−) = lim F (y )
y<x F (x+) = lim F (y ).
y>x Note that the CDF in Example 3.1.1 has six jumps of size 1/6. The jumps are located at
the six possible values of X , namely at the integers one through six, and the size of each of
those jumps is 1/6, which is the probability assigned to each of the six possible values. The
value of the CDF exactly at a jump point is equal to the right limit at that point. For example,
FX (1) = FX (1+) = 1/6 and FX (1−) = 0. The size of the jump at any x can be written as
FX (x) = FX (x) − FX (x−).
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This note was uploaded on 02/09/2014 for the course ISYE 2027 taught by Professor Zahrn during the Spring '08 term at Georgia Institute of Technology.
- Spring '08
- The Land