iscrete Uniform Distribution
المؤلف:
Sloane, N. J. A.
المصدر:
Sequences A086111 and A086112 in "The On-Line Encyclopedia of Integer Sequences."
الجزء والصفحة:
...
17-4-2021
2147
iscrete Uniform Distribution
The discrete uniform distribution is also known as the "equally likely outcomes" distribution. Letting a set 
have 
elements, each of them having the same probability, then
so using 
gives
 |
(5)
|
Restricting the set 
to the set of positive integers 1, 2, ..., 
, the probability distribution function and cumulative distributions function for this discrete uniform distribution are therefore
for 
, ..., 
.
The discrete uniform distribution is implemented in the Wolfram Language as DiscreteUniformDistribution[n].
Its moment-generating function is
The moments about 0 are
 |
(12)
|
so
and the moments about the mean are
The mean, variance, skewness, and kurtosis excess are
The mean deviation for a uniform distribution on 
elements is given by
 |
(24)
|
To do the sum, consider separately the cases of 
odd and 
even. For 
odd,
Similarly, for 
even,
The complete solution is therefore
![MD=<span style=]() {(N^2-1)/(4N) for N odd; 1/4N for N even. " src="https://mathworld.wolfram.com/images/equations/DiscreteUniformDistribution/NumberedEquation4.gif" style="height:72px; width:176px" /> |
(33)
|
For 
, 2, ..., the first few values are 0, 1/2, 2/3, 1, 6/5, 3/2, 12/7, ... (OEIS A086111 and A086112).
REFERENCES:
Sloane, N. J. A. Sequences A086111 and A086112 in "The On-Line Encyclopedia of Integer Sequences."
الاكثر قراءة في الاحتمالات و الاحصاء
اخر الاخبار
اخبار العتبة العباسية المقدسة
