Brainscape's Knowledge GenomeTM

Browse over 1 million classes created by top students, professors, publishers, and experts.


See full index

Data Science > Probability > Flashcards

Probability Flashcards

1
Q

Sample space and Events (Def)

A

The set of all possible outcomes in an experiment is called a SAMPLE SPACE (denoted S or Ω).

EVENTS are any possible subset of S

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
2
Q

Categories of Sample Spaces

A

There are 3 categories of Sample Spaces:

  • FINITE number of elements
  • INFINITE COUNTABLE
  • INFINITE UNCOUNTABLE
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
3
Q

Independent events (Def)

A

Two events E1 and E2 are independent if the outcome of E1 does not affect the outcome of E2, and viceversa

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
4
Q

Multiplication principle

A

Suppose we have n independent events E_1, E_2, … , E_n. If event E_k has m_k possible outcomes (for k = 1, 2, … , n), the there are

m_1 * m_2 * … * m_n

Possible ways for these events to occur

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
5
Q

k-Permutations w/o repetition

A

A way of selecting k objects from a list of n.

  • The order of selection matters
  • Each object can be selected only once
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
6
Q

k-Permutations w/ repetition

A

A way of selecting k objects from a list of n.

  • The order of selection matters
  • Each object can be selected more than once
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
7
Q

Combinations w/o repetition

A

A way of selecting k objects from a list of n.

  • The order of selection does NOT matter
  • Each object can be selected only once

Aka n-choose-k

This is also the BINOMIAL COEFFICIENT

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
8
Q

Combinations w/ repetition

A

A way of selecting k objects from a list of n.

  • The order of selection does NOT matter
  • Each object can be selected more than once

It’s aka the MULTISET COEFFICIENT

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
9
Q

Combinations w/ repetition (ice cream example)

A
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
10
Q

Definition of Probability

A

For a given experiment with Sample Space S, PROBABILITY is a real-valued function:

P: S -> [0, 1]

For each subset E that belongs to S, the function P assigns a number P(E), such that P(E) € [0, 1]

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
11
Q

Axioms of Probability

A

The re are 4 axioms:
(1) For any event E subset of S, 0 <= P(E) <= 1

(2) P(S) = 1
(3) For two disjoint events (such that their intersection = 0), P(EuF) = P(E) + P(F)
(4) More generally, (3) works for n mutually exclusive events too

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
12
Q

Conditional probability formula

A
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
13
Q

Mutual independence (Def)

A

Events A, B and C are MUTUALLY INDEPENDENT if:
- P(A&B&C) = P(A)P(B)P(C)
And
- Pairwise independent

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
14
Q

Law of Total Probability (Formula)

A

If {E_1, E_2, … , E_k} are a PARTITION of S, then for any event A:

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
15
Q

Bayes’ rule (Formula)

A
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
16
Q

Odds (Formula)

A

Historically, the likelihood of an event B has been expressed as a ratio between the prob of B and the prob of non-B

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
17
Q

Odds w/ Bayes’ Theorem

A
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
18
Q

Random Variable (Def)

A

A RANDOM VARIABLE is a function

X: S -> IR

For each element s of S, X(s) is a real number in IR

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
19
Q

Range space / Support (Def)

A

The RANGE SPACE (or Support) R_x of a random variable X is the set of all possible realisations of X

( X(s) for every s € S )

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
20
Q

Probability Mass Function (Def)

A

The PMF of a discrete random variable X is a function

f: R_x -> (0, 1]

Such that

f(x) = P(X=x) = p_x(x)

for each x € R_x such that

f(x) > 0
Σ_(x€R_x) f(x) = 1
P(X€A) = Σ_(x€A) f(x)

for some event A

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
21
Q

Expected value (Formula)

A
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
22
Q

Expected value (Formula)

A
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
23
Q

Variance (Formula)

A
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
24
Q

Cumulative Distribution Function (Formula)

A
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
25
Q

Bernoulli distribution (Def)

A

An experiment that can take two values, 1 (success) and 0 (failure), with

P(1) = θ
P(0) = 1-θ
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
26
Q

Binomial distribution (Def)

A

Repeated experiment n times, with each experiment a Bernoulli variable

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
27
Q

Poisson distribution (Def)

A

Describes the number of events occurring within a given interval, with rate λ.
It is commonly used to describe count data

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
28
Q

Joint PMF (Formula)

A
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
29
Q

Joint PMF (Key properties)

A
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
30
Q

Marginal distribution (Def and Formula)

A

Let X_i be the i-th component of a k-dimensional random vector X. The distribution function F_(X_i)(x) of X_i is called the MARGINAL DISTRIBUTION of X_i.

For a bivariate discrete r.v., it’s PMF is:

31
Q

Conditional PMF (Formula)

A
32
Q

Conditional vs. Marginal dist (Vis)

A
33
Q

Covariance (Formula)

A
34
Q

Covariance (Key properties)

A

(1) A positive value indicates a positive LINEAR relationship, and viceversa
(2) Zero indicates the variables are LINEARLY INDEPENDENT

Note:

35
Q

Correlation (Def and Formula)

A

Correlation is a measure of how strong the linear relationship is between two random variables

36
Q

Independence of r.v. (Def and Key properties)

A

Two r.v.s X and Y are independent if all events relating to X are independent of all events relating to Y.

The following statements are equivalent:
(1) X and Y are independent

(2) The JOINT PMF of X and Y is the product of the MARGINAL PMFs
(3) The CONDITIONAL distribution of X given Y=y does not depend on y, and viceversa

37
Q

Multinomial distribution (Def)

A

n independent trials with k possible outcomes for each trial. Each time, the probability of observing the j-th outcome is θ_j. Denote by X_j the number of times we observe the j-th outcome.

X = [ X_1, X_2, … , X_k]

X_1 + X_2 + … + X_k = n

θ_1 + θ_2 + … + θ_k = 1

38
Q

Multinoulli (n=1) distribution (E and Var)

A
39
Q

Multinomial distribution (Joint PMF)

A
40
Q

Multinomial distribution (E and Var)

A
41
Q

Multinoulli (n=1) distribution (Joint PMF)

A
42
Q

Multinoulli (n=1) distribution (Def)

A

Multinoulli is a multinomial distribution when n=1, i.e. there is only 1 trial, but still k possible outcomes

43
Q

Multinomial’s relationship to the Binomial dist

A

The Binomial is a special case of the Multinomial, where k=2 (i.e. only 2 possible outcomes).

If X ~ Binom(n, θ), then
X = (X, n-X) ~ Mu(n, (θ, 1-θ))

44
Q

The transformation theorem

A

Using the joint PMF/PDF it’s possible to find the expected value of any real function g(X, Y) of X and Y.

Let X, Y be a pair of discrete r.v. and g(X, Y) be any real-valued function of X and Y.
Then if it exists, the expected value of g(X, Y) is defined to be:

45
Q

Probability Density Function (Def)

A

For any a<=b, the probability P(a

46
Q

Continuous CDF (Formula)

A
47
Q

Normal distribution (Def)

A
48
Q

Calculating probabilities for the Normal distribution

A

You can calculate P(X<=x) in two stages, using the Standard Normal dist

Z ~ N(0, 1)

(1) Transform P(X<=x) into P(Z<=z)
(2) Use the CDF of Z to calculate probabilities

49
Q

Uniform distribution (Def)

A

X has a uniform distribution over the interval [a, b], written

X ~ U(a, b)

If it has PDF and CDF

50
Q

Uniform distribution (Var and E)

A
51
Q

Exponential distribution (Def)

A

X has an exponential dist with parameter λ>0, if it has PDF and CDF

52
Q

Exponential dist (Var and E)

A
53
Q

Joint PDF for bivariate continuous r.v. (PDF and Key properties)

A
54
Q

Marginal PDF for bivariate continuous r.v. (Formula)

A
55
Q

Marginal PDF for bivariate continuous r.v. (Examples)

A
56
Q

E and Var of a sum of r.v.

A
57
Q

Conditional dist of bivariate continuous r.v. (PDF)

A
58
Q

Conditional dist of bivariate continuous r.v. (Expected value)

A
59
Q

Law of Iterated Expectations

A
60
Q

Conditions for two r.v. To be independent

A

(1) f_XY(x, y) = f_X(x) * f_Y(y)
(2) The Joint PDF factorizes into:

f_XY(x, y) = C * g(x) * h(y)

With C some constant (the factorization is not unique)

(3) f_X|Y(x|y) = f_X(x) and viceversa

Conditions (1) and (2) require that the joint range space R_XY is the cartesian product of R_X and R_Y.

If (2) holds, then the Marginal PDFs of X and Y are proportional to g(x) and h(y), respectively

61
Q

Joint CDF for bivariate r.v. (Formula)

A
62
Q

Standard MVN - Multivariate Normal distribution (E and Var)

A
63
Q

MVN - Multivariate Normal distribution (E and Var)

A
64
Q

MVN (PDF)

A
65
Q

Marginal distributions of MVN

A
66
Q

Conditional distributions of MVN (Formula)

A
67
Q

Law of Large numbers (Def)

A
68
Q

Chebyshev’s WLLN

A
69
Q

Kolmogorov’s SLLN

A
70
Q

CLT (Def)

A

Let {X_n} be a sequence of r.v.s. Let X_n-bar be the sample mean of the first n terms of the sequence.

A CLT is a proposition giving a set of conditions to guarantee the convergence of the sample mean to a NORMAL DIST, as the sample size increases, i.e. sufficient to guarantee that

71
Q

CLT (Steps to use)

A

The CLT is used as follows:

(1) we observe a sample consisting of n observations X_1, X_2, … , X_n

(2) If n is large enough, then a standard normal distribution is a good approximation of the distribution of
sqrt(n) * (X_n-bar - μ) / σ

(3) Therefore, we pretend that

sqrt(n) * (X_n-bar - μ) / σ
~ N(0, 1)

(4) As a consequence, the distribution of the sample mean is

72
Q

CLT (Equivalent form)

A
73
Q

CLT - Normal approximation of the binomial

A

Let X_1, X_2, … , X_n be a sequence of iid Bernoulli(θ) r.v.s. We know that:

  • E[X_i] = θ and
    Var[X_i] = θ*(1-θ)
  • X = Σ(X_i) ~ Binom(n, θ)

The CLT tells us that:

74
Q

CLT - Normal approximation of the Poisson

A

Data Science (2 decks)

Brainscape helps you reach your goals faster, through stronger study habits.
© 2025 Bold Learning Solutions. Terms and Conditions