Math 2.0

Probability theory and stochastic processes Definition

Median

Builds on Random variableCumulative distribution function

Definition

Let XX be a real random variable on a probability space (Ω,ℱ,P)(\Omega, \mathcal{F}, P). A median of XX is a real number mm with P(X≤m)≥12andP(X≥m)≥12.P(X \le m) \ge \frac{1}{2} \quad \text{and} \quad P(X \ge m) \ge \frac{1}{2}.

Remarks

Each real random variable has a median, and the medians form a closed bounded interval. For example, Q(1/2)Q(1/2) is a median, where QQ is the quantile function. The median can be different from the mean. For the exponential distribution with rate λ\lambda, the median is ln⁡2/λ\ln 2 / \lambda and the mean is 1/λ1/\lambda. For a fair coin coded as 00 and 11, each m∈[0,1]m \in [0, 1] is a median. For a sorted data set x1≤⋯≤xnx_1 \le \dots \le x_n with nn odd, the median is x(n+1)/2x_{(n+1)/2}. For nn even, one usually takes the mean of the two middle values.

Reading path

To understand this page from the ground up, read these 41 pages in order. The foundations come first.

  1. Construction of the reals from Cauchy sequences concept
  2. Element definition
  3. Integers object
  4. Ordered field definition
  5. Proposition concept
  6. Random experiment concept
  7. Rational numbers object
  8. Real numbers object
  9. Set definition
  10. Ring of integers statement
  11. Disjunction definition
  12. Implication definition
  13. Predicate definition
  14. Ordered pair definition
  15. Union definition
  16. Sample space definition
  17. The rationals form a field statement
  18. Existential quantifier definition
  19. Universal quantifier definition
  20. Cartesian product definition
  21. Subset definition
  22. The rationals form an ordered field statement
  23. Uniqueness quantifier definition
  24. Binary relation definition
  25. Universal set definition
  26. Order on the reals definition
  27. Power set definition
  28. Complement definition
  29. Function definition
  30. Interval definition
  31. Preimage of a set definition
  32. Sigma-algebra definition
  33. Event definition
  34. Borel sigma-algebra definition
  35. Probability axioms concept
  36. Probability measure definition
  37. Probability distribution definition
  38. Probability space definition
  39. Random variable definition
  40. Cumulative distribution function definition
  41. Median definition

Where this leads

3 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Median, Probability distributions
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