CLT Cheat Sheet
For Sample Means of Samples of Size 𝒏
𝝁𝑿
𝝈𝑿
̅
Distribution of 𝒙
𝝁𝑿̅
𝜇𝑋 = 𝐸(𝑿)
𝜎𝑋 = 𝑆𝐷(𝑿)
𝑥̅ ~ normal(𝜇𝑋̅ , 𝜎𝑋̅ )
𝜇𝑋̅ = 𝐸(𝑿)
𝑿 ~ normal(𝜇, 𝜎)
𝜇𝑋 = 𝜇
𝜎𝑋 = 𝜎
𝑥̅ ~ normal(𝜇𝑋̅ , 𝜎𝑋̅ )
𝜇𝑋̅ = 𝜇
𝜎𝑋̅ =
𝑿 ~ exponential(𝛽)
𝜇𝑋 = 𝛽
𝜎𝑋 = 𝛽
𝑥̅ ~ normal(𝜇𝑋̅ , 𝜎𝑋̅ )
𝜇𝑋̅ = 𝛽
𝜎𝑋̅ =
𝑿 ~ Poisson(𝜆)
𝜇𝑋 = 𝜆
𝜎𝑋 = √𝜆
𝑥̅ ~ normal(𝜇𝑋̅ , 𝜎𝑋̅ )
𝜇𝑋̅ = 𝜆
𝜎𝑋̅ =
Distribution of 𝑿
(general)
𝑿 ~ a distribution
with expected value
E(X) and standard
deviation SD(X)
𝑿 ~ hyper.(𝑚, 𝑛, 𝑘)
𝜇𝑋 =
𝑘𝑚
𝑚+𝑛
𝑚
𝑛
𝑚+𝑛−𝑘
)(
)(
)
𝑚+𝑛 𝑚+𝑛 𝑚+𝑛−1
𝜎𝑋 = √𝑘 (
𝑥̅ ~ normal(𝜇𝑋̅ , 𝜎𝑋̅ )
𝜇𝑋̅ =
𝝈𝑿̅
𝑘𝑚
𝑚+𝑛
Distribution of 𝑿
𝝁𝑿
𝝈𝑿
̂
Distribution of 𝒑
𝝁𝒑̂
𝑿 ~ binomial(𝑛, 𝑝)
𝜇𝑋 = 𝑛𝑝
𝜎𝑋 = √𝑛𝑝(1 − 𝑝)
𝑝̂ ~ normal(𝜇𝑝̂ , 𝜎𝑝̂ )
𝜇𝑝̂ = 𝑝
𝜎𝑋̅ =
𝑆𝐷(𝑿)
𝜎𝑋̅ =
√𝑛
𝜎
√𝑛
𝛽
√𝑛
√𝜆
√𝑛
𝜎𝑋
√𝑛
𝝈𝒑̂
𝜎𝑝̂ =
√𝑝(1 − 𝑝)
√𝑛
Note: in the “4.1 Video 6 (CLT Cheat Sheet)” video, the final row shows a distribution for 𝑿 ~ binomial(1, 𝑝) instead of 𝑿 ~ binomial(𝑛, 𝑝), but that ended
up being confusing for a lot of people, so I’ve updated this sheet to show information for the distribution for 𝑿 ~ binomial(𝑛, 𝑝).
© 2018 Claudia Mahler