Exponential (Gamma) Distributions (SOA Exam P – Probability – Univariate Random Variables)

Описание: Master the Exponential Distribution and its Gamma Distribution generalization for SOA Exam P. In this lesson you’ll learn how to recognize exponential/gamma PDFs at a glance, parameterizations (θ vs λ), key properties (support, expected value, variance), the gamma function intuition, when an exponential is just a special case of gamma, plus a quick example and a faster “tabular” trick for repeated integration by parts. Perfect for actuarial candidates who want exam-day speed and accuracy.

What you’ll learn
• Identify exponential vs gamma from the density’s shape
• Map parameters and switch between θ and λ
• Compute mean/variance quickly from parameters
• Use proportionality to spot gamma (no heavy calculus)
• Apply a tabular IBP shortcut on exam-style integrals

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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.

Gamma Distribution: 𝑋 ~ Γ(𝛼,𝜃)

Supp(𝑋)=(0,∞)
𝑓_𝑋 (𝑥) ∝ 𝑥^(𝛼−1)∙𝑒^((−𝑥)/𝜃)

𝐸[𝑋]=𝛼∙𝜃
𝑉𝑎𝑟(𝑋)=𝛼∙𝜃^2

Exponential Distribution: 𝑋 ~ 𝐸𝑥𝑝(𝜃)=Γ(𝛼=1,𝜃)
Supp(𝑋)=(0,∞)
𝑓_𝑋 (𝑥)=1/𝜃 𝑒^((−𝑥)/𝜃)
𝐹_𝑋 (𝑥)=1−𝑒^((−𝑥)/𝜃)

𝐸[𝑋]=𝜃
𝑉𝑎𝑟(𝑋)=𝜃^2
𝑆_𝑋 (𝑥)=𝑒^((−𝑥)/𝜃)

Example:

A random variable 𝑋 has probability density function 𝑓_𝑋 (𝑥)=125/2 𝑥^2 𝑒^(−5𝑥) for 𝑥 lager than 0. Determine the mean of the random variable.
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