Probability distribution : PMF vs CDF vs PDF

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Probability mass function vs probability density function 

• PMF (Probability Mass Function): This is used for discrete random variables. The PMF gives the probability that a discrete random variable takes on a specific value. For example, if you're rolling a six-sided die, the PMF would tell you the probability of rolling any particular number (e.g., $P(X = 3) = \frac{1}{6}$).

• PDF (Probability Density Function): This is used for continuous random variables. Unlike the PMF, the PDF does not give the probability of the random variable taking on a specific value (since that probability is technically 0 for continuous variables). Instead, the PDF describes the relative likelihood that the random variable falls within a particular range of values. The area under the PDF curve over an interval gives the probability that the random variable falls within that interval. https://medium.com/probablity-and-statistics-for-data-science/probability-distributions-5f457cde025e

  
  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  In the picture, Y-axis in the normal distribution represents the "density of probability." Intuitively, it shows the chance of obtaining values near corresponding points on the X-axis.

  The area under the curve between two X values tells the probability (or percent frequency) of variables taking on a range of the values.

The Cumulative Distribution Function (CDF) of a random variable X, denoted as F(x), is defined as the probability that X takes on a value less than or equal to x.

$F(x)=P(X\le x)$

This function gives us the cumulative probability up to a certain point x.

In summary:

• PMF is for discrete variables and provides the probability of exact outcomes.
• PDF is for continuous variables and provides a density function, where probabilities are found by integrating over an interval.