Beta Distributions (SOA Exam P – Probability – Univariate Random Variables)
Автор: AnalystPrep
Загружено: 2023-02-04
Просмотров: 12315
Master the Beta Distribution for SOA Exam P. In this lesson you learn what the Beta distribution is, how to recognize it from a proportional density, why its support is between 0 and 1, and how to get the mean, second raw moment, and variance without grinding through long integrals. We finish with a full worked example so you can compute variance quickly and confidently on the exam.
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After completing this video you should be able to:
Explain and calculate expected value and higher moments, mode, median, and percentile.
Beta Distribution: 𝑋 ~ 𝛽(𝑎,𝑏)
𝑓_𝑋 (𝑥)=(Γ(𝑎+𝑏))/(Γ(𝑎)∙Γ(𝑏))∙𝑥^(𝑎−1)∙(1−𝑥)^(𝑏−1)
𝐸[𝑋]=𝑎/(𝑎+𝑏)
𝑉𝑎𝑟(𝑋)=𝐸[𝑋^2 ]−(𝐸[𝑋])^2
Example:
A random variable 𝑋 has probability density function 𝑓_𝑋 (𝑥)=60𝑥^3 (1−𝑥)^2 for 𝑥 between 0 and 1. Determine the variance of the random variable.
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