Please help on arithmetic mean of continuous distributions.

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SUMMARY

The arithmetic mean (X bar) of a continuous distribution is defined by the integral formula ∫x.f(x)dx, where 'a' is the lower limit and 'b' is the upper limit of integration. To establish this, one should first prove the discrete case of the expectation theorem before applying integral calculus principles. Understanding the integral as a limiting sum is crucial for grasping the transition from discrete to continuous distributions. Engaging with the proof process is essential for effective learning and comprehension.

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  • Integral calculus fundamentals
  • Understanding of discrete probability distributions
  • Familiarity with expectation theorems
  • Basic knowledge of continuous probability distributions
NEXT STEPS
  • Study the proof of the expectation theorem for discrete distributions
  • Learn about the properties of integrals in calculus
  • Explore the concept of limiting sums in calculus
  • Investigate applications of the arithmetic mean in continuous distributions
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Students of mathematics, statisticians, and anyone interested in probability theory and the application of calculus in understanding continuous distributions.

AAQIB IQBAL
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PROVE mean (X bar) of a continuous distribution is given by:

∫x.f(x)dx
{'a' is the lower limit of integration and 'b' is the upper limit}
 
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AAQIB IQBAL said:
PROVE mean (X bar) of a continuous distribution is given by:

∫x.f(x)dx
{'a' is the lower limit of integration and 'b' is the upper limit}


Hello AAQIB and welcome to the forums.

I think the best way would be to first prove the case for the discrete case and then use integral definitions in calculus to prove the integral way.

If you haven't taken a thorough course in calculus, you'll find that the integral is simply a special kind of limiting sum, just like the differential is a quantity that is calculated through a limiting argument.

Note as well that we can't just give you the answer, we ask that you show your working so we can provide hints and let you attempt to figure it out so that you learn for yourself.

So in this spirit, can you first prove the discrete version of the expectation theorem?
 

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