Math-Stat L1

Continuous Variables

• Definition: Can take any value within a range, represented by a continuum (real numbers, including fractions and decimals).

Probability in Continuous Variables

• Point Probability: P(X = x) is always 0.

• Probability Density Function (PDF): Distributes probability across intervals; calculated via integration:

  • P(a < X < b) = ∫[a, b] p(x) dx.

Properties of Density Function

• Range of Densities: Values are positive and total area under PDF equals 1.

• Support:

  • Unbounded: Covers entire real line (e.g., Normal distribution).

  • Semi-infinite: Limited to positive values (e.g., Exponential distribution).

  • Bounded: Finite interval [a, b].

Mean and Expected Value

• Mean (E[X]): E[X] = ∫[support] x * p(x) dx; reflects the distribution's symmetry.

Cumulative Distribution Function (CDF)

• CDF Definition: Probability that x ≤ a value, represented as F(x) or P(X ≤ x).

• Derivation: F(x) = ∫[min, x] p(t) dt.

Transformations of Random Variables

• Overview: If y has a distribution and x is a transformation of y, derive x's CDF through:

  • P(X ≤ x) = P(g(Y) ≤ x).

Example: Uniform Distribution

• The CDF on [0, 1] has constant density of 1.

Conclusion

Understanding continuous variables involves probabilities, transformations, means, and CDFs, all influenced by distribution properties.