Continuous Random Variables Summary

Probability · Axiom Academy

SUMMARY Continuous Random Variables Let's review the key concepts, formulas, and techniques for working with continuous random variables. PDF (Probability Density Function): Describes the density of probability at each point. Must satisfy and CDF (Cumulative Distribution Function): Gives , obtained by integrating the PDF Key Relationship: The PDF is the derivative of the CDF: Point Probabilities: For continuous RVs, (probabilities only exist for intervals) Key Differences from Discrete RVs Probability Calculation: Use integrals instead of sums. PDF Values: Can exceed 1 (unlike PMF). The PDF measures density, not probability Expectation: Replace summation with integration: Total Probability: The area under the PDF curve equals 1, representing certainty Example Recap: Computing Expectation and Variance Step 1 - Verify the PDF: Confirm that and the PDF integrates to 1 Step 2 - Compute Expectation: Calculate using integration by parts or other techniques Step 3 - Find Second Moment: Compute to prepare for variance calculation Step 4 - Calculate Variance: Use to find the spread of the distribution Important Integral Formulas and Techniques Law of Unconscious Statistician: Joint PDF: describes the probability density for two variables simultaneously Marginal PDFs: Obtained by integrating out the other variable. For X: Independence: X and Y are independent if and only if for all x, y Conditional PDF: gives the distribution of X given Y = y

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