ECO 045 EXAM 2 Study Guide PDF

Preview

Summary

ECO 045 EXAM 2 STUDY GUIDE2. Probability Theory ● Experiment - a random process that generates well defined outcomes ● Sample point - one possible experimental outcome ○ ex. 6 of spades ● Sample space(s) - collection of all possible experimental outcomes ○ ex. all 52 cards ● Event (A, B, etc.) - any...

Description

ECO 045 EXAM 2 STUDY GUIDE 2. Probability Theory ● Experiment - a random process that generates well defined outcomes ● Sample point - one possible experimental outcome ○ ex. 6 of spades ● Sample space(s) - collection of all possible experimental outcomes ○ ex. all 52 cards ● Event (A, B, etc.) - any combination of outcomes, any subset of the sample space ○ ex. drawing a face card, 12 sample points ■ Intersection (𝐴 ∩ 𝐵 ) - sample points in event A and B ■ Union (𝐴 ∪ 𝐵 ) - sample points in event A or B 𝑐

■ Complement (𝐴 ) - sample points not in event A Basic rules of probability ● Positivity - the probability of any single outcome is between 0 and 1 ● Completeness - the sum of the probabilities of all possible outcomes is equal to 1 ● Additivity - for mutually exclusive events, 𝐴 , 𝐴 ,... , 𝐴 , the probability of union is sum of 1

2

𝑛

individual probabilities: 𝑃(𝐴 𝑈𝐴 𝑈... 𝑈 𝐴 )= Σ𝑃(𝐴 ) 1

●

2

𝑛

𝑖

○ Mutually exclusive - events A and B have no sample points in common Joint and conditional probabilities ○ Marginal probability - the probability of an event occurring, thought of as an unconditional probability ○ Joint probability - probability of the intersection of two events, often defined for individual sample points ■ ○

𝑃(9 𝑜𝑓 ℎ𝑒𝑎𝑟𝑡𝑠)= 𝑃(9 ∩ ℎ𝑒𝑎𝑟𝑡𝑠) =

1 52

Conditional probability - probability of one event, given that another event definitely occurred

■ 𝑃(𝐴|𝐵) =

𝑃(𝐴∩𝐵) 𝑃(𝐵)

●

Independent and dependent events ○ Independent events - conditional probabilities are equal to marginal probabilities ■ 𝑃(𝐴|𝐵) = 𝑃(𝐴) ■ 𝑃(𝐵|𝐴) = 𝑃(𝐵) ○ Dependent events - a relationship exists between the events ■ 𝑃(𝐴|𝐵) ≠ 𝑃(𝐴) General Properties of Probability ● Addition Law - union of events ○ 𝑃(𝐴 ∪ 𝐵) = 𝑃(𝐴) + 𝑃(𝐵) − 𝑃(𝐴 ∩ 𝐵) ○ For mutually exclusive events: 𝑃(𝐴 ∪ 𝐵) = 𝑃(𝐴) + 𝑃(𝐵) ○

𝑐

For complements: 𝑃(𝐴) = 1 − 𝑃(𝐴 ) 𝑐

■ 𝐴 + 𝐴 is mutually exclusive 𝑐

●

●

𝑐

● 𝑃(𝐴 ∪ 𝐴 ) = 𝑃(𝐴) + 𝑃(𝐴 ) = 1 Multiplication Law - intersection of events ○ For dependent events: 𝑃(𝐴 ∩ 𝐵) = 𝑃(𝐴)𝑃(𝐵|𝐴) ○ For independent events: 𝑃(𝐴 ∩ 𝐵) = 𝑃(𝐴)𝑃(𝐴) Bayes’ Theorem

𝑃(𝐴)𝑃(𝐵|𝐴)

○ 𝑃(𝐴|𝐵) =

𝑃(𝐵)

𝑃(𝐴)𝑃(𝐵|𝐴) 𝑐 𝑐 𝑃(𝐴)𝑃(𝐵|𝐴)+𝑃(𝐴 )𝑃(𝐵|𝐴 )

=

●

Steps to solve probability problems: 1. Write down the probabilities given, using the notation P(A), P(A|B) 2. Define the probability you want, in terms of the notation 3. Find the appropriate property 4. Plug in and solve 3. Discrete Probability Distribution ● Random variable - a variable that takes numeric values as the result of a random process or experiment ○ Converts random outcome to numeric value ■ ex. dots on upward face of die: number of dots ■ ex. People who show up for a flight: number of people ○ Probability functions require a numeric “x” ● Mean (“expected value”) of probability distribution: µ = Σ 𝑥 𝑓 (𝑥) ● ●

2

2

Variance of probability distribution: σ = Σ(𝑥 − µ) 𝑓 (𝑥) Combination - number of different possible combinations of x elements that can be made out of a larger set of n elements (order does not matter) 𝑛 𝑛! ○ (𝑥 ) = 𝑥!(𝑛−𝑥)!

○ 𝑛𝐶𝑥 ●

●

Probability Function - formula for the probability of each possible value of a random variable ○ Corresponds to a particular probability distribution (binomial, uniform, etc) ○ 𝑓(𝑥) : probability the random variable has a value equal to x ○ Requirements: ■ 𝑓(𝑥) ≥ 0 (positivity) ■ Σ𝑓(𝑥) = 1 (completeness) ○ 𝑃(𝑥 > # 𝑦) = 𝑓(0) + 𝑓(1) +... + 𝑓(𝑦) (“cumulative” property) ○ 𝑃(𝑥 ≤ # 𝑦 ) = 1 − [𝑓(𝑦 + 1) + 𝑓(𝑦 + 2) +... + 𝑓(𝑛)] Binomial distribution - describes probability of x “successes” out of n independent “trials” where the probability of success in each trial is P ○ 𝑥~𝑏𝑖𝑛(𝑛, 𝑝) 𝑛

𝑥

(𝑛−𝑥)

○ 𝑃(𝑥) = 𝑓(𝑥) = (𝑥)𝑝 (1 − 𝑝)

●

○ Mean: 𝑛𝑝 ○ Variance: 𝑛𝑝(1 − 𝑝) Discrete Uniform Probability Function - random variable x takes values from 1 to n with equal probability for each value ○ ex. roll 1 die (n=6) ○ ex. draw 1 card (n=52) ○

𝑓(𝑥) =

1 𝑛

○

Mean:

𝑛+1 2

○

Variance:

(for 𝑥 = 1, 2,... , 𝑛)

2

(𝑛 −1) 12

Continuous Probability Distribution ● Random variable x takes any value in a continuous range: 𝑥 ∈ [𝑚𝑖𝑛(𝑥), 𝑚𝑎𝑥(𝑥)] ○ ex. 𝑥 ∈ (− ∞, + ∞) ● Probability that x equals some value exactly is zero: 𝑃(𝑥 = 𝑎) = 0 ● Instead of𝑃(𝑥 = 𝑎) exactly, the probability that x falls within some interval can be computed ○ 𝑃(𝑎 ≤ 𝑥 ≤ 𝑏) ○ Use a p.d.f., 𝑓(𝑥) ○ Probability of an interval is the area under the curve of that interval → rectangles ■ ○

𝑃(𝑎 ≤ 𝑥 ≤ 𝑏) = (𝑏 − 𝑎) × 𝑓(𝑥) = (𝑤𝑖𝑑𝑡ℎ) × (ℎ𝑒𝑖𝑔ℎ𝑡) = (𝑏 − 𝑎) ×

=

𝑏−𝑎 𝐵−𝐴

Area under curve: 𝑏

■

𝑏

𝑃(𝑎 ≤ 𝑥 ≤ 𝑏) = ∫ 𝑓(𝑥)𝑑𝑥 = ∫ 𝑎

●

1 𝐵−𝐴

𝑎

1 𝐵−𝐴

𝑑𝑥

Continuous Uniform Distribution - random variable x takes any value in some range [A,B] with uniform probability ○

P.d.f.: 𝑓(𝑥) =

1 𝐵−𝐴

, 𝑖𝑓 𝑥 ∈ [𝐴, 𝐵]

Normal Distribution ● ●

The “bell curve”, crucial for statistical inference - describes sampling distribution of averages (𝑥) Normal density function: ○ Mean: µ ○ Standard deviation: σ

○ ●

P.d.f.:

𝑓(𝑥) =

1 σ 2π

𝑒

−

(𝑥−µ) 2σ

2

2

Probability of Ranges in Standard Normal Distribution: ○

Below some value: 𝑃(−

∞ < 𝑧 ≤ 𝑧𝑈) = 𝐹(𝑧𝑈)

○

Above some value: 𝑃(𝑧

≤ 𝑧...