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/actuarial-exams-complete-courses-by-analystprep/ After completing this video you should be able to: - Explain and calculate expected value and higher moments, mode, median, and percentile. Gamma Distribution: ? ~ Γ(?,?) Supp(?)=(0,∞) ?_? (?) ∝ ?^(?−1)∙?^((−?)/?) ?[?]=?∙? ???(?)=?∙?^2 Exponential Distribution: ? ~ ???(?)=Γ(?=1,?) Supp(?)=(0,∞) ?_? (?)=1/? ?^((−?)/?) ?_? (?)=1−?^((−?)/?) ?[?]=? ???(?)=?^2 ?_? (?)=?^((−?)/?) Example: A random variable ? has probability density function ?_? (?)=125/2 ?^2 ?^(−5?) for ? lager than 0. Determine the mean of the random variable.
AnalystPrep Actuarial Exams Study Packages (video lessons, study notes, question bank, and quizzes) can be found at https://analystprep.com/shop/actuarial-exams-complete-courses-by-analystprep/ After completing this video you should be able to: - Explain and calculate expected value and higher moments, mode, median, and percentile. Gamma Distribution: ? ~ Γ(?,?) Supp(?)=(0,∞) ?_? (?) ∝ ?^(?−1)∙?^((−?)/?) ?[?]=?∙? ???(?)=?∙?^2 Exponential Distribution: ? ~ ???(?)=Γ(?=1,?) Supp(?)=(0,∞) ?_? (?)=1/? ?^((−?)/?) ?_? (?)=1−?^((−?)/?) ?[?]=? ???(?)=?^2 ?_? (?)=?^((−?)/?) Example: A random variable ? has probability density function ?_? (?)=125/2 ?^2 ?^(−5?) for ? lager than 0. Determine the mean of the random variable.