How can I calculate the PMF from a given CDF?

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SUMMARY

The discussion focuses on calculating the Probability Mass Function (PMF) from a given Cumulative Distribution Function (CDF). The CDF provided is defined piecewise, with specific values at intervals: 0 for x ≤ -1, 0.2 for -1 ≤ x < 0, 0.7 for 0 ≤ x < 1, and 1 for x ≥ 1. The PMF can be derived using the formula f(x) = ΔF(x)/Δx, which represents the change in the CDF over the change in x. This method allows for the determination of the probabilities associated with discrete outcomes based on the provided CDF.

PREREQUISITES
  • Understanding of Cumulative Distribution Functions (CDF)
  • Knowledge of Probability Mass Functions (PMF)
  • Familiarity with piecewise functions
  • Basic calculus concepts, specifically differentiation
NEXT STEPS
  • Study the relationship between CDF and PMF in discrete probability distributions
  • Learn how to derive PMF from various types of CDFs
  • Explore piecewise function definitions and their applications in probability
  • Review basic calculus, focusing on differentiation techniques relevant to probability
USEFUL FOR

Statisticians, data scientists, and students studying probability theory who need to understand the conversion between CDF and PMF for discrete random variables.

EugP
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How do I find PMF if I have CDF? For example:

[tex] F_X (x)=\left\{\begin{array}{cc}0,&\mbox{ if }x\leq -1<br /> \\0.2, & \mbox{ if } -1\leq x < 0<br /> \\0.7, & \mbox{ if } 0\leq x < 1<br /> \\1, & \mbox{ if } x \geq 1<br /> \end{array}\right.[/tex]
 
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f(x) = [itex]\Delta F(x)/\Delta x[/itex]
 

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