STATS

Conditional Probability Formula

P(A|B) = P(A∩B) / P(B)

Baye’s Theorem

P(A|B) = [ P(B|A) * P(A) ] / P(B)

Set Theory: Union

Elements in all of A or B : (A∪B)

Set Theory: Intersection

Elements in only A and B : (A∩B)

Set Theory: Complement

Elements not within a specific set: (A' or A^c)

Set Theory: Relative Complement

Elements in a set and not in any other : (A - B or A \ B)

Set Theory: Symmetric Difference

Elements in either set but NOT both : (A Δ B)

Circular Permutation Formula

(n-1)!

When can you do n^k?

1. There are exactly k distinct slots

2. Independent choices/same number of options each slot

3. Repetition MUST BE ALLOWED

Combinations with repetition (Stars and Bars)

C( n + k - 1, k - 1) Where n is total options and k is groups.

Combinations Formula

C(n, r) = n! / [ (n-r)! r! ]

Permutations Formula

P(n, r) = n! / [ (n-r)! ]

Probability Mass Function (PMF)

The probability of a single, exact value occurring for a discrete random variable →(P(X=x))

Cumulative Distribution Function (CDF)

The running total (accumulated probability) of a random variable being less than or equal to a specific value ((P(X <= k)).

Binomial Distribution Formula

“Chance of x successes out of n fixed trials?” P(X = x) = f(x) = C(n, x) * pˣ * (1−p)ⁿ⁻ˣ

n = # of trials

x = # of “successful” outcomes

p = probability of a single “success”

1-p = probability of “failure”

Probability Density Function (PDF)

A mathematical function used for continuous random variables

where the area under the curve represents the probability of a range of outcomes.

Going from PDF (point) to F (accumulator) uses…

Integration 3.17

Going from F (accumulator) to PDF (point) uses…

Differentiation 3.69

How to get marginal density

If you want x then you integrate over all possible values of y and treat the other variable(s) as constants. 3.40

Expected Value

E(X) = Σ[ x * f(x) ] (Multiply each value by its probability then add them all)

2 Events are independent if…

P(A∩B) = P(A) * P(B)

2 Events are independent if…

P(A | B) = P(A)

Variance

How spread out values are from the average. Var(X) = σ² = E[ (X-μ)² ] OR E[X²] - (E[X])²

Expected Value

E(X) = ∫ [ x * f(x) ] from its bounds. f(x) is marginal density of x

Linear Changes of E[X]

E[aX + b] = a*E[X] + b

Correlation Coefficient

ρ = [ Cov(X,Y) ] / [σ_x * σ_y]

Covariance

E[XY] - E[X] * E[Y]

E[XY]

Double integral of xy * joint distribution

Negative Binomial

Independent trials, and you’re counting trials until a success. C(x−1, k−1) pᵏ (1−p)ˣ⁻ᵏ