Distribution of a random variable , pdf vs probability distribution

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SUMMARY

The discussion clarifies the relationship between the distribution of a random variable and probability distributions, emphasizing that a probability distribution describes the range of values a random variable can take and their associated probabilities. It highlights the distinction between probability mass functions (PMFs) and probability density functions (PDFs), noting that while both provide probabilities, a probability distribution function encompasses a broader concept. Additionally, it is established that a probability distribution function may exist without a corresponding density function, as it can be non-differentiable.

PREREQUISITES
  • Understanding of random variables
  • Familiarity with probability mass functions (PMFs)
  • Knowledge of probability density functions (PDFs)
  • Basic concepts of differentiability in mathematics
NEXT STEPS
  • Study the properties of probability mass functions (PMFs) in discrete distributions
  • Explore the characteristics of probability density functions (PDFs) in continuous distributions
  • Learn about the implications of differentiability in probability distribution functions
  • Investigate real-world applications of random variables and their distributions
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Students, statisticians, and data analysts seeking to deepen their understanding of probability theory and its applications in various fields.

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Hey all i struggling to understand, these concepts. would some explain to me the relationship and differences the distribution of a random variable and a probabiltiy distribution.
wikipedia says this about probability distribution "The probability distribution describes the range of possible values that a random variable can attain and the probability that the value of the random variable is within any (measurable) subset of that range."

Now if you have probability mass funtion or density funtion which gives you the probability of a given set. why do we need a probability distribution funtion what does it buy us, it seems like they both giving us the same things

any help would me much appreciated
 
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It is possible to have a probability distribution function which is not differentiable, so there will be no density function.
 

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