scholarly journals On the ubiquitous notion of mean in probability and statistics

2019 ◽  
Vol 2 (2) ◽  
pp. 35-41
Author(s):  
Aboubakar MAITOURNAM
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Law Of Large Numbers ◽  
Probability Distributions ◽  
Basic Notion ◽  
The Core ◽  
The Central Limit Theorem ◽  
Large Numbers ◽  
Essential Parameter ◽  
Probability And Statistics

In probability and statistics, the basic notion of probability of an event can be expressed as a mathematical expectation. The latter is a theoretical mean and is an essential parameter of most probability distributions, in particular of the Gaussian distribution. Last but not least, the notion of mean is at the core of two main theorems of probabilities and statistics, that is : the law of large numbers and the central limit theorem. Whether it is a theoretical or empirical version, the concept of mean is omnipresent in probability and statistics, is consubstantial to these two disciplines and is a bridge between randomness and determinism.

Random splittings of an interval

1993 ◽  
Vol 30 (01) ◽  
pp. 131-152
Author(s):  
T. S. Mountford ◽  
S. C. Port
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Law Of Large Numbers ◽  
Unit Interval ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Strong Law ◽  
The Central Limit Theorem ◽  
Large Numbers ◽  
Necessary And Sufficient

Points are independently and uniformly distributed onto the unit interval. The first n—1 points subdivide the interval into n subintervals. For 1 we find a necessary and sufficient condition on {ln } for the events [Xn belongs to the ln th largest subinterval] to occur infinitely often or finitely often with probability 1. We also determine when the weak and strong laws of large numbers hold for the length of the ln th largest subinterval. The strong law of large numbers and the central limit theorem are shown to be valid for the number of times by time n the events [Xr belongs to l r th largest subinterval] occur when these events occur infinitely often.

Central Limit Theorem for Time to Broadcast in Radio Networks

1995 ◽  
Vol 9 (2) ◽  
pp. 201-209 ◽  
Author(s):  
Krishnamurthi Ravishankar ◽  
Suresh Singh
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Law Of Large Numbers ◽  
Radio Networks ◽  
The Other ◽  
The Central Limit Theorem ◽  
Large Numbers ◽  
Constant Radius ◽  
Radio Transmitters

We study the problem of broadcasting in a system where nodes are equipped with radio transmitters with constant radius of transmission. A message originating at a node has to be transmitted to all the other nodes in the system. We prove the central limit theorem and the law of large numbers for the number of time steps required to complete a broadcast for the case when the nodes are placed on a line independently uniformly distributed. We show that the number of time steps required to broadcast is 3n/4 in probability.

Random splittings of an interval

1993 ◽  
Vol 30 (1) ◽  
pp. 131-152
Author(s):  
T. S. Mountford ◽  
S. C. Port
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Law Of Large Numbers ◽  
Unit Interval ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Strong Law ◽  
The Central Limit Theorem ◽  
Large Numbers ◽  
Necessary And Sufficient

Points are independently and uniformly distributed onto the unit interval. The first n—1 points subdivide the interval into n subintervals. For 1 we find a necessary and sufficient condition on {ln} for the events [Xn belongs to the ln th largest subinterval] to occur infinitely often or finitely often with probability 1. We also determine when the weak and strong laws of large numbers hold for the length of the ln th largest subinterval. The strong law of large numbers and the central limit theorem are shown to be valid for the number of times by time n the events [Xr belongs to lr th largest subinterval] occur when these events occur infinitely often.

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