STATISTICS FOR E&BE | Summary (RUG)

Statistics for E&BE
(University of Groningen)
Summary 2022-2023
Stuvia: marcellaschrijver




Ch 4. Probability: The Study of Randomness ........................................................................................ 2
Ch 5. Random Variables and Probability Distributions ......................................................................... 4
Ch 6. Sampling Distributions .................................................................................................................. 8
Ch 7. Introduction to Inference.............................................................................................................. 9
Ch 8. Inference for the Mean of a Population ..................................................................................... 11
Ch 10. Inference for Proportions ......................................................................................................... 14
Ch 11. Inference for Categorical Data .................................................................................................. 15

,Ch 4. Probability: The Study of Randomness
Randomness
Uncertain individual outcomes regularly distributed in a large number of repetitions

Probability
Proportion of times that an outcome occurs in many repeated trials
𝑃(∅) = 0
0 ≤ 𝑃(𝐴) ≤ 1
𝑃(𝑆) = 1
𝑃(𝐴𝑐 ) = 1 − 𝑃(𝐴)

Equally likely outcomes
|𝐴| number of outcomes in 𝐴
𝑃(𝐴) = |𝑆|
= number of outcomes in 𝑆


Addition rule
𝑃(⋃𝑛𝑖=1 𝐴𝑖 ) = ∑𝑖 𝑃(𝐴𝑖 ) − ∑𝑖<𝑗 𝑃(𝐴𝑖 ∩ 𝐴𝑗 ) + ∑𝑖<𝑗<𝑘 𝑃(𝐴𝑖 ∩ 𝐴𝑗 ∩ 𝐴𝑘 ) − ⋯ + (−1)𝑛+1 𝑃(𝐴𝑖 ∩ … ∩

Conditional probability
Probability of an event differs if we know that another event occurred
𝑃(𝐴 ∩ 𝐵)
𝑃(𝐴|𝐵) =
𝑃(𝐵)

Multiplication rule
𝑃(𝐴 ∩ 𝐵) = 𝑃(𝐴|𝐵)𝑃(𝐵) = 𝑃(𝐵|𝐴)𝑃(𝐴)
𝑃(⋂𝑛𝑖=1 𝐴𝑖 ) = 𝑃(𝐴1 )𝑃(𝐴2 |𝐴1 )𝑃(𝐴3 |𝐴1 , 𝐴2 ) … 𝑃(𝐴𝑛 |𝐴1 , … , 𝐴𝑛−1 )

Independent events
Outcome isn’t affected by outcome of another trial
𝑃(𝐴|𝐵) = 𝑃(𝐴)
Never when sampling without replacement

Multiplication rule
𝑃(𝐴 ∩ 𝐵) = 𝑃(𝐴|𝐵)𝑃(𝐵) = 𝑃(𝐴)𝑃(𝐵)
𝑃(𝐴 ∩ 𝐶) = 𝑃(𝐴|𝐶)𝑃(𝐶) = 𝑃(𝐴)𝑃(𝐶)
𝑃(𝐵 ∩ 𝐶) = 𝑃(𝐵|𝐶)𝑃(𝐶) = 𝑃(𝐵)𝑃(𝐶)
𝑃(𝐴 ∩ 𝐵 ∩ 𝐶) = 𝑃(𝐴)𝑃(𝐵)𝑃(𝐶)

Disjoint events
Cannot occur simultaneously
𝐴𝑖 ∩ 𝐴𝑗 = ∅ for 𝑖 ≠ 𝑗
Never independent

Addition rule
𝑃(⋃∞ ∞
𝑛=1 𝐴𝑛 ) = ∑𝑛=1 𝑃(𝐴𝑛 )
Law of total probability (LOTP)
𝑃(𝐴) = ∑𝑛𝑖=1 𝑃(𝐴|𝐵𝑖 )𝑃(𝐵𝑖 ) where 𝐵1 , … , 𝐵𝑛 is a partition of 𝑆

, Probability tree




Bayes’ rule
𝑃(𝐵|𝐴)𝑃(𝐴) 𝑃(𝐵|𝐴)𝑃(𝐴)
𝑃(𝐴|𝐵) = 𝑐 )𝑃(𝐴𝑐 ) =
𝑃(𝐵|𝐴)𝑃(𝐴) + 𝑃(𝐵|𝐴 𝑃(𝐵)