Exponential (Gamma) 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.
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.

[@sintumavuya7495]

I couldn't for the life of me understand how people were seeing Gamma or Beta functions from a pdf. You have just saved my life with these 2 videos on those 2. You're my hero today. Thank you.

[@DirBirder]

I paid for lessons and just could not understand the Gamma Distributions nor the example problems given with them. This video made it click for me on the first watch. I'll be watching the whole playlist now, I absolutely love the quality of explanation and teaching here. Thank you so much doctor. I'm very happy to really understand Gamma problems now.

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[@lukascho6297]

This series is amazing

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Thank you. Your playlist has been very helpful

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[@hitm43]

Amazing repeated integration by parts trick. I've never seen that.

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[@18softball33]

This may be late in the series to be asking this- but why do we use brackets around X for the expected value, and parentheses for the variance?

[@lukascho6297]

It's just a notation convention.

[@Johann01198]

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