Exponential (Gamma) Distributions (SOA Exam P – Probability – Univariate Random Variables)
AnalystPrep Actuarial Exams Study Packages (video lessons, study notes, question bank, and quizzes) can be found at https://analystprep.com/shop/actuaria...
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.
