Continuous probability distribution

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SUMMARY

Continuous probability distributions are essential in various fields, particularly in modeling real-world phenomena. They describe random variables that can take on any value within a range, as opposed to discrete distributions that only take specific values. The normal distribution, a key example, is widely utilized in sciences such as physics, psychology, and business for statistical analysis. Practical applications include digital transmission systems, where continuous random variables represent noise and signal interference, impacting data recovery and error rates.

PREREQUISITES
  • Understanding of random variables and their classifications (discrete vs. continuous)
  • Familiarity with normal distribution and its properties
  • Basic knowledge of statistical analysis techniques
  • Insight into digital communication systems and signal processing
NEXT STEPS
  • Explore the properties and applications of the normal distribution in various fields
  • Learn about statistical analysis methods for minimizing bit error rates in digital communication
  • Investigate the role of continuous random variables in queuing theory and event timing
  • Study the implications of continuous probability distributions in machine learning algorithms
USEFUL FOR

Data scientists, statisticians, engineers, and professionals in fields requiring statistical modeling and analysis, particularly in digital communications and behavioral sciences.

JamieLam
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Hi, I'm not sure if this has been brought up before. I'm a non-mathematician. I like to know what's the use of continuous probability distribution. Is there any use for it, is it merely a mathematical object or has it real(practical uses for it) If there are practical uses for it, what is it been use for? Thank you very much
 
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JamieLam said:
Hi, I'm not sure if this has been brought up before. I'm a non-mathematician. I like to know what's the use of continuous probability distribution. Is there any use for it, is it merely a mathematical object or has it real(practical uses for it) If there are practical uses for it, what is it been use for? Thank you very much

Wellcome on MHB JamieLam!...

Basically there are two types of random variables. One type is a r.v. that assume a discrete set of values [i.e. the result of coin toss where, for example, You consider r=1 the head and r=-1 the tail...] and they are described by a discrete probability function. Another type of r.v. can assume a continous set of values [i.e. the temperature of a room that can assume any value of $\displaystyle T_{min} \le T \le T_{max}$...] and they are described by a continous probability function... Kind regards $\chi$ $\sigma$
 
JamieLam said:
Hi, I'm not sure if this has been brought up before. I'm a non-mathematician. I like to know what's the use of continuous probability distribution. Is there any use for it, is it merely a mathematical object or has it real(practical uses for it) If there are practical uses for it, what is it been use for? Thank you very much

One of the continuous distributions is the so called normal distribution, which is shaped like a bell curve.
The normal distribution is extensively applied in many, many sciences, including physics, psychology, and business.
 
I like Serena said:
One of the continuous distributions is the so called normal distribution, which is shaped like a bell curve.
The normal distribution is extensively applied in many, many sciences, including physics, psychology, and business.

Thank you Chisigma and I like Serena for the warm welcome and kind guidance, for r.v. that has a discrete set of values, I understand that for real life examples are like dice and coins. Is there any uses for r.v. that uses continuous probability function for real life? I mean I assume there are but I personally do not know any. If you know, please say. Thanks!
 
A suggestive example comes from my experience in the field of digital transmission, i.e. the medium that supports for You Internet, Smartphone, GPS and other modern services. The transmitted signal s(t) is a sequence of symbols, one any T seconds and the sequence is recovered sampling in appropiate way the received signal any T seconds. The received signal r(t) is the sum of an highly attenuated reply of the transmitted signal and thermal noise of the electonic circuits of the reciever so that can be written as...

[FONT=MathJax_Math-italic][FONT=ea9bd3dac1f0b279081a2160#081300]r[FONT=MathJax_Main][FONT=7253d757afc94e7c081a2160#081300]([FONT=MathJax_Math-italic][FONT=ea9bd3dac1f0b279081a2160#081300]t[FONT=7253d757afc94e7c081a2160#081300][FONT=MathJax_Main])[FONT=MathJax_Main]=[FONT=MathJax_Math-italic][FONT=ea9bd3dac1f0b279081a2160#081300]a[FONT=MathJax_Math-italic][FONT=ea9bd3dac1f0b279081a2160#081300]s[FONT=MathJax_Main][FONT=7253d757afc94e7c081a2160#081300]([FONT=MathJax_Math-italic][FONT=ea9bd3dac1f0b279081a2160#081300]t[FONT=7253d757afc94e7c081a2160#081300][FONT=MathJax_Main])[FONT=MathJax_Main]+[FONT=MathJax_Math-italic][FONT=ea9bd3dac1f0b279081a2160#081300]n[FONT=MathJax_Main][FONT=7253d757afc94e7c081a2160#081300]([FONT=MathJax_Math-italic][FONT=ea9bd3dac1f0b279081a2160#081300]t[FONT=7253d757afc94e7c081a2160#081300][FONT=MathJax_Main])[FONT=MathJax_Main]([FONT=MathJax_Main]1[FONT=MathJax_Main])[FONT=7253d757afc94e7c081a2160#081300]

Now n(t) is a continuos r.v. that is described in term of continuous probability function. If for example thye tramsitted signal is a sequence of binary symbols, each with possible values + 1 or - 1, if in the sampling time t the istantaneous value of the noise n(t) overcomes s(t), then You have an erroneous recovered symbol, i.e. a trasmitted 1 is sampled as -1 or vice versa. In the figure You can see a binary received signal corrupted by noise...

http://ddpozwy746ijz.cloudfront.net/c1/78/i83392705._szw380h285_.jpg

A very important design target in a radio or optical digital receiver is to minimize the bit error rate and an essential role to meet that is the statistical analysis of noise resistance of the receiver...

Kind regards

[FONT=ea9bd3dac1f0b279081a2160#081300][FONT=MathJax_Math-italic]χ [FONT=MathJax_Math-italic][FONT=ea9bd3dac1f0b279081a2160#081300]σ[FONT=ea9bd3dac1f0b279081a2160#081300]
 
chisigma said:
A suggestive example comes from my experience in the field of digital transmission, i.e. the medium that supports for You Internet, Smartphone, GPS and other modern services. The transmitted signal s(t) is a sequence of symbols, one any T seconds and the sequence is recovered sampling in appropiate way the received signal any T seconds. The received signal r(t) is the sum of an highly attenuated reply of the transmitted signal and thermal noise of the electonic circuits of the reciever so that can be written as...

[FONT=MathJax_Math-italic][FONT=ea9bd3dac1f0b279081a2160#081300]r[FONT=MathJax_Main][FONT=7253d757afc94e7c081a2160#081300]([FONT=MathJax_Math-italic][FONT=ea9bd3dac1f0b279081a2160#081300]t[FONT=7253d757afc94e7c081a2160#081300][FONT=MathJax_Main])[FONT=MathJax_Main]=[FONT=MathJax_Math-italic][FONT=ea9bd3dac1f0b279081a2160#081300]a[FONT=MathJax_Math-italic][FONT=ea9bd3dac1f0b279081a2160#081300]s[FONT=MathJax_Main][FONT=7253d757afc94e7c081a2160#081300]([FONT=MathJax_Math-italic][FONT=ea9bd3dac1f0b279081a2160#081300]t[FONT=7253d757afc94e7c081a2160#081300][FONT=MathJax_Main])[FONT=MathJax_Main]+[FONT=MathJax_Math-italic][FONT=ea9bd3dac1f0b279081a2160#081300]n[FONT=MathJax_Main][FONT=7253d757afc94e7c081a2160#081300]([FONT=MathJax_Math-italic][FONT=ea9bd3dac1f0b279081a2160#081300]t[FONT=7253d757afc94e7c081a2160#081300][FONT=MathJax_Main])[FONT=MathJax_Main]([FONT=MathJax_Main]1[FONT=MathJax_Main])[FONT=7253d757afc94e7c081a2160#081300]

Now n(t) is a continuos r.v. that is described in term of continuous probability function. If for example thye tramsitted signal is a sequence of binary symbols, each with possible values + 1 or - 1, if in the sampling time t the istantaneous value of the noise n(t) overcomes s(t), then You have an erroneous recovered symbol, i.e. a trasmitted 1 is sampled as -1 or vice versa. In the figure You can see a binary received signal corrupted by noise...

http://ddpozwy746ijz.cloudfront.net/c1/78/i83392705._szw380h285_.jpg

A very important design target in a radio or optical digital receiver is to minimize the bit error rate and an essential role to meet that is the statistical analysis of noise resistance of the receiver...

Kind regards

[FONT=ea9bd3dac1f0b279081a2160#081300][FONT=MathJax_Math-italic]χ [FONT=MathJax_Math-italic][FONT=ea9bd3dac1f0b279081a2160#081300]σ[FONT=ea9bd3dac1f0b279081a2160#081300]

Thanks for the information. If you have more examples of the use of continuous probability functions used in stuff that laymen commonly used, please say. :)
 
The amount of time between two events is often a continuous random variable.

eg the number of seconds elapsed between two busses arriving at the same bus stop could be:
0.0000000000000000000000000000000000000000001
0.000000000000000000000000000002
7.1
or any other arbitrary number
 

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