Beta Distributions (SOA Exam P - Probability - Univariate Random Variables)

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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.

[@aria_nukil]

This was a fun explanation. I wonder if this distribution is used to model the binary outcomes similar to how a logistic regression is used. Thanks

[@yaweli2968]

Why not memorize the var(x) instead of E(x^2). I believe you want to do the least work as possible on the exam.

[@fahadnizamani2487]

I applied the formula of variance of beta distn of kind 1.. but the answer was different.. and also the var of beta distn of 2nd kind has different variance.. both variances do not match with Professor's answer.. i dont know why!