Let be a positive integer and let be a continuous random variable that is uniformly distributed on . For any number , denote by the largest integer not exceeding . Similarly, denote to be the fractional part of . The following are two properties of and :
1. For any and , we have:
.
This property follows directly from the definition of the fractional part function: is the difference between and the largest integer not exceeding it, so it follows that the sum of the integer part and the fractional part of is equal to .
2. For any , we have:
.
This property is a consequence of the fact that is uniformly distributed on , meaning that for any number , the probability that falls within the interval is equal to the length of that interval divided by the total length of . Since the total length of is , the probability that falls within the interval is .
These properties can be useful in analyzing and deriving properties of random variables that are uniformly distributed on different intervals.