Econ 41: Key Probability Terms & Definitions Study Set

Distributive Laws

A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

De Morgan's Laws

(A ∪ B)' = A' ∩ B' and (A ∩ B)' = A' ∪ B' (Signs Swap)

mutually exclusive events

events have no intersection, cannot both occur A1 ∩ A2 = Ø

nPr

n!/(n-r)! Used with n different objects, select r and order them

nCr

n!/r!(n-r)! = nC(n-r) Used with n different objects, choose r no order

formula for non-distinct objects

nCr Used with n objects of 2 types, r of type 1, n-r of type 2

0!

1

nC0

nCn-0 = nCn = 1

nC1

nCn-1 = n

Conditional Probability Formula

P(A|B) = P(A ∩ B) / P(B)

Complement Rule of Conditional Probability

P(B'|A) = 1 - P(B|A)

Independent Events

P (A ∩ B) = P(A) x P(B), if new information is given and it doesn't update the probability of an event occurring, P(B|A) = P(B)

Independence vs. Mutually Exclusive Events

If A and B are independent, then they cannot be mutually exclusive

If A and B are mutually exclusive, then they cannot be independent

Complements of Independent Events

If A and B are independent

A and B' are independent

A' and B are independent

A' and B' are independent

Pairwise Independence

for a triplet, can be pairwise independent but not mutually independent

Occurs when events

A and B are independent

B and C are independent

A and C are independent

Mutual Independence

A, B, and C are mutually independent if they are pairwise independent

AND P(A ∩ B ∩ C) = P(A) x P(B) x P(C)

Complements of Independent Events

If A, B, and C are mutually independent then all combinations of them and their complements will also be mutually independent

what to do with questions about independence

use intersections somehow to be able to multiply probabilities

Addition Law

P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

Properties of PMF

if f(x) is the pmf than it is greater than or equal to 0

the Sigma of all x within S = 1

P(X within A) = Sigma of x within (S ∩ A)

CDF

cumulative distribution function

F(x) = P(X less than or equal to x) for any x within all real numbers

mean

average value of X, sigma of all x within S (xf(x)) - multiply each x by f(x) and add them together

variance

sigma of all x within S (x-mean)^2 f(x)- x minus the mean squared times f(x) OR x squared times f(x) then subtract the mean squared

Expectation

if X has a pmf f(x) then for any function u(x), the expectation of u(x) is sigma u(x) f(x) - or u(x) times f(x) added for all values of x

Expectation in regards to x

the expectation when u(x) = x, is sigma f(x) x - or the mean, f(x) times x

Expectations with constants

if k is a constant then E [k] = k, if E[k u(x)]= k E[u(x)]

Linearity of Expectations

E [k1 u(x) + k2 u2(x)] = k1 E[u(x)] + k2 E[u2(x)]

mean and variance when Y = aX + B

mean of Y = a(mean of X) + b ||||| variance of Y = a^2 (variance of X)

Binomial Random Variable

trial with 2 outcomes (success and failure), repeated independently - called a Bernoulli trial

p represents the probability of success in the n number of trials

X represents number of successes in the n number of trials

f(x) = nCx p^x (1-p)^(n-x) for x=0,1,2...

Mean and variance of binomials

if X~ (n,p) then the mean = np, and variance = np(1-p)

X ~ b (n,p) meaning

b - undichtes binomial

n - number of trials

p - probability of success in each trial

Linearity of Binomials

if X1 ~ b (n,p) and X2 = n-X1 then X2 ~ b(n, 1-p)