Continuous Probability Distributions

Introduction to CDF and PDF

• Continuous Random Variable (RV): Represented by the variable x.
• Cumulative Distribution Function (CDF): Denoted as CDF(x), gives the probability that the random variable X is less than or equal to x.
• Probability Density Function (PDF): Denoted as f(x). Obtained by taking the derivative of the CDF.
  • Relationship: If CDF(x) is the cumulative function, then
    $f(x) = \frac{d}{dx} (CDF(x))$
  • Importance: Essential to switch from CDF to PDF for continuous distributions.

Checking CDF for a Function of x

• Function g(x): If given a function g of the random variable X, the task is to derive the CDF for g(X).
• Finding the CDF:
  • The CDF of g(X) involves determining the CDF of f(X) and substituting it appropriately based on the transformation defined by g.
  • Resulting function is significant and must be handled with care.

Expectation of Continuous Random Variables

• Definition of Expected Value:
  • For a continuous random variable X, the expected value (mean) is calculated using the PDF.
  • The formula is
    $E[X] = ext{Integral of } x imes f(x) ext{ over the support}$
• Integration Over Support: Commonly, the support is from negative infinity to positive infinity, but it may be restricted sometimes.
  • Value of Density: Density can be zero outside the support range.
• Variable Transformation:
  • One can change the variable in the integration and denote it as t, leading to adjustments in the integral.
• Example: The Expected Value Calculation
  • Consider a function where the PDF has a range, such as [0,1].
  • If f(x) = 2x, then
    $E[X] = ext{Integral of } x imes 2x dx ext{ from 0 to 1}$
  • Carrying out the integration,
  • Result: $E[X] = 2 imes \frac{x^3}{3} \bigg|_0^1 = \frac{2}{3}$

Variance Calculation for Continuous Random Variables

• Definition of Variance: Variance quantifies the spread of a random variable. Denoted as $Var(X) = E[X^2] - (E[X])^2$
• Importance: Measures how the density spreads out, indicating whether the probability is concentrated (less variance) or spread out (greater variance).
• Example Calculation:
  • Given density. Calculate $E[X^2]$ as follows:
  1. Find the PDF.
  2. Compute $E[X^2] = ext{Integral of } x^2 imes f(x) ext{ over the support}$
  3. Then use the variance formula: $$ Var(X) = E[X^2] - (E[X])^2 $$
  4. For example, integrating $x^2$ where $f(x) = 2x$ between [0,1].

Applications of Uniform Distribution

• Uniform Random Variable:
  • A uniform distribution is one where all outcomes are equally likely.
  • The general form of a uniform distribution on [a, b] is known and the density function is given as
    $f(x) = \begin{cases} \frac{1}{b-a} & \text{if } a <= x <= b \ 0 & \text{otherwise} \end{cases}$
• CDF of Uniform Distribution:
  • The CDF, which shows the integral of the PDF, behaves in a piecewise manner:
  • For x < a: CDF(x) = 0
  • For x between [a,b]: CDF(x) = $\frac{x-a}{b-a}$
  • For x > b: CDF(x) = 1

Generating Random Variables using Uniform Distribution

• Generating Variable Y from U:
  • To generate a new random variable Y using a uniform distribution:
  1. Generate a uniform random variable U (e.g., U ~ Uniform(0,1)).
  2. Find the CDF of Y, then take the inverse to determine the new variable Y as
     $Y = F^{-1}(U)$
• Inverse Function Application: Essential for obtaining new random variables that follow different distributions (like exponential).

Further Considerations

• Statistical Analysis Tools: R and other statistical software can streamline these processes, especially with simulation or numerical methods.
• Integration of CDFs and PDFs: Comprehension: Understanding all steps leading from PDF to CDF will enrich grasping continuous probability across various mathematical models.

Conclusion

• This document summarizes the mathematical concepts surrounding continuous random variables, including their expected values and variances, while demonstrating how to compute relevant statistics through derivations from the CDF to the PDF. Emphasis is on the ability to manipulate these distributions to find new random variables through transformations and use of software.