Things to Know 1 Midterm

What are the 2 rules for a partition of S?

1. S1+S2+…+Sn = S

2. Si ∩ Sj = null set for all i ≠ j (aka pairwise disjoint)

What are the 3 axioms of probability?

1. Nonnegativity → P(A) >= 0 for all A

2. Additivity → for disjoint: P(U_i^k A) = sum for all i of P(A_i)

3. Normalization → P(omega) = 1

What’s the multiplication rule? What a good way around it?

P(A∩B) = P(A|B)P(B) = P(B|A)P(A)

What does the law of total probability say?

For Si partitioning S, P(A)

= P(A∩S1) + P(A∩S2) + … + P(A∩Sn)

= P(A|S1)P(S1) + … + P(A|Sn)P(Sn)

What does P(A|B) = 0.4 tell you?

P(A^c|B) = 0.6 (NOT P(A|B^c))

What’s the independence formula for 2 events? Conditionally independent? RVs?

P(A∩B) = P(A)P(B) → independent (works for more than 2 events)

P((A∩B)|C) = P(A|C)P(B|C) → conditionally independent

P(X=x, Y=y) = P(X=x)P(Y=y) → independent RVs

Does pairwise independence imply independence of all events?

NO

What is P(A) for uniform?

P(A) = |A| / |S|

What is the k permutation formula?

n! / (n-k)!

Order matters

What is the combination formula?

n! / (n-k)!k!

Remove all the ways that are the same but different orderings (order doesn’t matter)

What is the multinomial formula?

n! / n1!n2!…nk!

How many ways to split n items into groups of size n1, n2, …, nk

For binomial Bin(n,p), list what X is, values it takes, a rule that must be true, variables and what they are, PMF, E(X), Var(X),

X = # successes in n trials

x = 0,1,2,…, n

independent and identically distributed (iid)

n = # trials, p = P(success)

P(X = k) = (n choose k)p^k(1-p)^n-k

E(X) = np

Var(X) = np(1-p)

For uniform Uniform(a,b), list what values X takes, what a and b are, PMF, E(X), Var(X)

x = a, a+1, a+2, …, b

a = lower bound, b = upper bound (inclusive)

pX(x) = 1/(b-a+1)

E(X) = (b-a)/2

Var(X) = [(b-a+1)²-1] / 12

For geometric Geo(p), list what X is, values it takes, a rule that must be true, variables and what they are, PMF, E(X), Var(X)

X = # trials until first success

x = 1,2,…

independent and identically distributed (iid)

p = P(success)

pX(k) = (1-p)^k-1p

E(X) = 1/p

Var(X) = (1-p)/p²

For poisson Poisson(λ), list what X is, values it takes, variables and what they are, PMF, E(X), Var(X)

X = # occurrences in an interval

x = 0,1,2,…

λ = rate (avg per interval)

pX(k) = e^-λ * (λ^k / k!)

E(X) = λ

Var(X) = λ

What is an atomic event?

1 outcome (can’t be decomposed)

What is a RV really?

Real valued function (map from sample space to numerical value)

What are the 3 rules of Probability Mass Functions (PMF)?

Same as axioms of probability

1. Nonnegativity → pX(x) >= 0 for all x

2. Additivity → P(X=x1 or X=x2) = P(X=x1) + P(X = x2)

3. Normalization → sum of all x of pX(x) = 1

What is the formula for expected value discrete? Continuous? Joint discrete?

sum of all x of xpX(x)

∫ -infinity to infinity of xpX(x)

sum of all x and all y of xyp_X,Y(x,y)

What is the formula for variance?

E(X²) - E(X)²

What is the nth moment generally?

E(Xⁿ)

Expectation: first moment

Variance: second moment - first moment²

What is standard deviation and what’s always true?

sqrt(Var(X))

Always exists and is nonnegative

What are the properties of variance?

Var(cX) = c²Var(X)

Var(k) = k

Var(aX + bY) = a²Var(X) + b²Var(Y) + 2abCov(X,Y)

What is the geometric series? When is it correct?

Sum from k=0 to infinity of ar^k = a / (1 - r)

Only true when |r| < 1

How do you find the marginal of x and y from a joint pmf?

pX(x) = sum of all y of P(X = x, Y = y) aka all the values of y in this x

pY(y) = sum of all x of P(X = x, Y = y) aka all the values of x in this y

What is conditional probability for multivar? What independence formula does this give you?

P(X=x | Y=y) = P(X=x, Y=y) / P(Y=y)

P(X=x | Y=y) = P(X=x) → independent

What does independence mean for covariance?

Cov(X,Y) = 0

What are the properties of expectation?

E(k) = k

E(kX) = kE(X)

E(X+Y) = E(X) + E(Y)

E(XY) = E(X)E(Y) + Cov(X,Y)

E(E(X|Y)) = E(X)

For continuous RV, what is P(X = x)?

0

What are the 2 rules of Probability Density Functions (PDFs)?

1. Nonnegativity → fX(x) >= 0 for all x

2. Normalization → integral from -infinity to infinity of fX(x)dx = 1

List what the following integrals are equal to: integral from a to b of fX(x)dx, ∫ Uⁿdu, ∫ 1/u du, ∫ e^audu, ∫a^udu, d/dx(sinx), d/dx(cosx), d/dx(cscx), d/dx(secx), d/dx(tanx), d/dx(cotx), sin2theta, cos2theta, d/dx(lnx), integration by parts, ∫ a to b + ∫ b to c

FX(b) - FX(a)

Uⁿ⁺¹/ n+1 + C

ln|u| + C

(1/a)e^au + C

a^u / ln(a) + C

cosx

-sinx

-cscxcotx

secxtanx

sec²x

-csc²x

2sinthetacostheta

2cos²theta - 1 = 1 - 2sin²theta = cos²theta - sin²theta

1/x

∫udv = uv - ∫vdu

∫ a to c

How do you find the cumulative distribution function continuous? What is it equivalent to? How do you find the CDF from the PDF? What is a property of CDFs?

∫ -infinity to x of fX(t)dt

P(X <= x) = FX(x)

d/dx(FX(x)) = fX(x) (PDF slope of CDF)

Monotonically nondecreasing

How is the normal distribution often notated? What is E(X) and Var(X)?

X ~ N(mew, sigma²)

E(X) = mew, Var(X) = sigma²

What does the Central Limit Theorem (CLT) say?

Sum of large # of independent and identically distributed (iid) RVs has an approximately normal behavior

What is the standard normal? How do you standardize a non-standard normal RV?

Z ~ N(0,1)

P(a < Y < b) → P(a-mew_Y / sigma_Y < Z < b-mew_Y / sigma_Y)