Introductory Probability and Set Theory Flashcards

Flashcards covering introductory set theory, probability rules, combinatorics, random variables, expectation, variance, and probability distributions based on lecture notes.

What is a set and how is its cardinality defined?

A set is any well-defined collection of objects or items, with members referred to as elements. The cardinality of a set $A$, denoted by $n(A)$, is the total number of elements contained in set $A$.

How are the union and intersection of two sets defined?

The union of two sets $A$ and $B$, denoted by $A \cup B$, contains all outcomes belonging to $A$ alone, $B$ alone, or both. The intersection, denoted by $A \cap B$, contains only outcomes that belong to both $A$ and $B$.

What is a power set and what defines disjoint sets?

A power set $P(A)$ is the set of all possible subsets of a given set $A$, including the empty set. Two sets $A$ and $B$ are disjoint if they share no elements in common, meaning $A \cap B = \phi$.

What are De Morgan's Laws in set theory?

De Morgan's Laws state that $(A \cup B)^c = A^c \cap B^c$ and $(A \cap B)^c = A^c \cup B^c$.

How is the Cartesian product of two sets $A$ and $B$ defined?

The Cartesian product $A \times B$ is the set of all possible ordered pairs $(x_1, x_2)$ where $x_1 \in A$ and $x_2 \in B$, written as $A \times B = \{(x_1, x_2) \mid x_1 \in A, x_2 \in B\}$.

What is the difference between mutually exclusive and independent events?

Two events $A$ and $B$ are mutually exclusive if both cannot happen simultaneously, meaning $A \cap B = \phi$. They are independent if the occurrence of one event does not affect the occurrence of the other in any trial, meaning $P(A \cap B) = P(A) \cdot P(B)$.

What is Stirling's Approximation formula for large factorials?

Stirling's formula approximates $n!$ when $n$ is large as $n! \approx \sqrt{2 \pi n} \, n^n e^{-n}$, where $e = 2.71828$ and $\pi = \frac{22}{7}$.

What are the formulas for arranging $k$ objects from $n$ distinct objects with and without replacement?

With replacement, the number of arrangements is $n^k$. Without replacement, the number of $k$-permutations is $^n P_k = \frac{n!}{(n-k)!}$.

What is the formula for permutations when some objects are identical?

If among $n$ objects, $n_1$ are identical, $n_2$ are identical, …, and $n_k$ are identical such that $\sum n_i = n$, the number of distinct arrangements is $\frac{n!}{n_1! n_2! \dots n_k!}$.

How is a combination defined mathematically?

A combination defines the number of ways to select $k$ objects from $n$ distinct objects without regard to order, calculated as $^n C_k = \binom{n}{k} = \frac{n!}{(n-k)! k!}$.

What is the statement of the Binomial Theorem?

For any two real numbers $x$ and $y$ and any positive integer $n$, $(x+y)^n = \sum_{k=0}^n \binom{n}{k} x^{n-k} y^k$.

What are the Classical and Relative Frequency definitions of probability?

The Classical definition states that if an experiment has $N$ equally likely outcomes and event $A$ contains $n(A)$ outcomes, $P(A) = \frac{n(A)}{N}$. The Relative Frequency definition states that if an experiment is performed $n$ times under identical conditions and event $A$ occurs $m$ times, $P(A) = \frac{m}{n}$.

What properties must a probability measure satisfy under the Axiomatic Definition?

1. $0 \le P(A) \le 1$\n2. $P(S) = 1$ and $P(\phi) = 0$\n3. For any sequence of mutually exclusive events $A_1, A_2, \dots$, $P(\bigcup_{i=1}^{\infty} A_i) = \sum_{i=1}^{\infty} P(A_i)$.

What is the Additive Rule of probability for non-mutually exclusive events?

For any two non-mutually exclusive events $A$ and $B$, the probability that $A$ or $B$ occurs is $P(A \cup B) = P(A) + P(B) - P(A \cap B)$.

How is conditional probability defined?

The conditional probability of event $A$ given that event $B$ has occurred is $P(A|B) = \frac{P(A \cap B)}{P(B)}$, provided that $P(B) > 0$. If $A$ and $B$ are independent, $P(A|B) = P(A)$.

What are the formulas for Total Probability and Bayes' Theorem?

Given $m$ mutually exclusive and collectively exhaustive events $A_1, A_2, \dots, A_m$, Total Probability is $P(B) = \sum_{i=1}^m P(A_i) \cdot P(B|A_i)$. Bayes' Formula for $P(A_i|B)$ is $P(A_i|B) = \frac{P(A_i) \cdot P(B|A_i)}{\sum_{j=1}^m P(A_j) \cdot P(B|A_j)}$.

What conditions must a discrete Probability Mass Function (PMF) satisfy?

A discrete PMF $p(x)$ must satisfy:\n1. $p(x_i) \ge 0$ for all $x_i$\n2. $\sum p(x_i) = 1$

How are expectation and variance calculated for a discrete random variable?

The mean (expectation) is $E(X) = \sum x p(x)$. The variance is $Var(X) = E(X^2) - [E(X)]^2$, where $E(X^2) = \sum x^2 p(x)$.

What are the algebraic properties of variance?

For constant $C$ and random variable $X$:\n1. $Var(C) = 0$\n2. $Var(X + C) = Var(X)$\n3. $Var(C X) = C^2 Var(X)$\n4. $Var(X_1 + X_2) = Var(X_1) + Var(X_2)$ if $X_1$ and $X_2$ are independent.

What conditions must a continuous Probability Density Function (PDF) satisfy?

A continuous PDF $f(x)$ must satisfy $f(x) \ge 0$ for all $x$ and $\int_{-\infty}^{\infty} f(x) \, dx = 1$.

What are the PMF, mean, variance, and MGF of a Bernoulli distribution?

For $X \sim \text{Bernoulli}(p)$ with $x \in \{0, 1\}$ and $q = 1 - p$:\n- PMF: $P(X=x) = p^x q^{1-x}$\n- Mean: $E(X) = p$\n- Variance: $Var(X) = pq$\n- MGF: $M_X(t) = p e^t + q$

What are the PMF, mean, variance, and MGF of a Binomial distribution?

For $X \sim b(n, p)$ with $x \in \{0, 1, 2, \dots, n\}$ and $q = 1 - p$:\n- PMF: $P(X=x) = \binom{n}{x} p^x q^{n-x}$\n- Mean: $E(X) = np$\n- Variance: $Var(X) = npq$\n- MGF: $M_X(t) = (p e^t + q)^n$

What are the PMF, mean, variance, and MGF of a Poisson distribution?

For $X \sim P(\lambda)$ with $x \in \{0, 1, 2, \dots\}$:\n- PMF: $P(X=x) = \frac{e^{-\lambda} \lambda^x}{x!}$\n- Mean: $E(X) = \lambda$\n- Variance: $Var(X) = \lambda$\n- MGF: $M_X(t) = e^{\lambda(e^t - 1)}$

What are the PMF, mean, and variance of a Geometric distribution for the number of failures before the 1st success?

For $X \sim G(p)$ representing failures before the 1st success ($x \in \{0, 1, 2, \dots\}$):\n- PMF: $P(X=x) = p q^x$\n- Mean: $E(X) = \frac{q}{p}$\n- Variance: $Var(X) = \frac{q}{p^2}$

What are the PMF, mean, and variance of a Hypergeometric distribution?

For $X \sim HG(N, a, n)$ representing number of defective items in sample size $n$ chosen without replacement from population $N$ with $a$ defectives:\n- PMF: $P(X=x) = \frac{\binom{a}{x}\binom{N-a}{n-x}}{\binom{N}{n}}$\n- Mean: $E(X) = n p$ where $p = \frac{a}{N}$\n- Variance: $Var(X) = n p q \left(\frac{N-n}{N-1}\right)$ where $q = 1 - p$

Under what condition can a Binomial distribution be approximated by a Poisson distribution?

When the number of trials $n$ is large and the probability of success $p$ is small, the Binomial distribution $b(n, p)$ can be approximated by a Poisson distribution $P(\lambda)$ with parameter $\lambda = np$.

Under what condition can a Hypergeometric distribution be approximated by a Binomial distribution?

When the total population size $N$ is very large, the Hypergeometric distribution $H(x; N, a, n)$ tends toward a Binomial distribution $b(n, p)$, where $p = \frac{a}{N}$.