Comprehensive Guide to Chance Processes and Distributions

Probability

The mathematics of chance, describing long-term patterns of unpredictable events.

Random Process

A situation in which we know the possible outcomes but cannot predict the specific outcome.

Law of Large Numbers (LLN)

States that as the number of trials increases, the observed frequency of an event approaches its true probability.

Independence

Condition where the outcome of one trial does not affect the outcome of another.

Complement ($A^C$)

The event that $A$ does not occur; calculated as $P(A^C) = 1 - P(A)$.

Mutually Exclusive

Two events that cannot happen at the same time; defined by $P(A ext{ and } B) = 0$.

Independent Events

Events where the probability of one occurring does not affect the other; defined by $P(A|B) = P(A)$.

General Addition Rule

The rule for finding the probability of the union of two events: $P(A ext{ or } B) = P(A) + P(B) - P(A ext{ and } B)$.

General Multiplication Rule

The rule for intersection probability: $P(A ext{ and } B) = P(A) imes P(B|A)$.

Conditional Probability

The probability of event $A$ occurring given that $B$ has already occurred, calculated as $P(A|B) = rac{P(A ext{ and } B)}{P(B)}$.

Sample Space ($S$)

The set of all possible outcomes of an experiment.

Event

A subset of outcomes from the sample space.

Variance

A measure of how far a set of numbers is spread out from their mean.

Standard Deviation

The square root of the variance, indicating the typical distance of outcomes from the mean.

Discrete Random Variable

A random variable that can take on a countable number of distinct values.

Expected Value (Mean)

The average or mean of a discrete random variable, found using $etaX = ext{sum of } (xi imes p_i)$.

Binomial Distribution

A probability distribution that summarizes the likelihood of a value taking on two possible outcomes, given a fixed number of trials.

Geometric Distribution

A probability distribution that models the number of trials until the first success occurs.

P(X=k)

The probability of exactly $k$ successes in $n$ trials for a binomial distribution.

Mean of Binomial Distribution

Calculated as $ ext{mean} = np$, where $n$ is the number of trials and $p$ is the probability of success.

Standard Deviation of Binomial Distribution

Calculated as $ ext{sd} = rac{ ext{sqrt}(np(1-p))}$.

Total Probability

The sum of the probabilities of all of the outcomes associated with a random variable.

Tree Diagrams

A visual representation used to organize and calculate probabilities of sequences of events.

Probability of Success

In binomial or geometric distributions, the likelihood of achieving success in a single trial.

Probability of Failure

Calculated as $1 - p$ in a binomial or geometric distribution context.

Expected Wait Time (Geometric)

The average number of trials needed for one success, calculated as $rac{1}{p}$.

Linear Transformations of Random Variables

If $Y = a + bX$, then $ ext{mean}Y = a + b* ext{mean}X$ and $ ext{sd}Y = |b| ext{sd}X$.

Sums of Random Variables

If $X$ and $Y$ are added, their means are added: $ ext{mean}{X+Y} = ext{mean}X + ext{mean}_Y$.

Differences of Random Variables

If $X$ is subtracted from $Y$, their means are also subtracted: $ ext{mean}{X-Y} = ext{mean}X - ext{mean}_Y$.

Variance for Independent Variables

For independent variables, variances are added: $ ext{variance}{X+Y} = ext{variance}X + ext{variance}_Y$.

Common Pitfall: Adding Standard Deviations

Never add standard deviations. Always use variances.

Binomial Conditions (BINS)

Binary, Independent, Number of Trials fixed, Success with constant probability.

Geometric Conditions (BITS)

Binary, Independent, Trials leading to the first success, Success with constant probability.

Misconception: Law of Averages

Short-run regularities do not exist; past independent outcomes do not affect future probabilities.

Complementary Events

If $A$ is an event, then the probability of its complement is $P(A^C) = 1 - P(A)$.

Threshold for Large Sample Size

As sample sizes increase, the relative frequency approaches theoretical probability.

Probability of At Least One Success

For geometric distribution, can be calculated as $1 - P( ext{No Successes})$.

Probability Mass Function (PMF)

A function that gives the probability of each value of a discrete random variable.

Cumulative Distribution Function (CDF)

A function that shows the probability that a random variable is less than or equal to a certain value.

Independence Check

To determine if two events are independent, check if $P(A|B) = P(A)$ holds true.

Statistical Independence

The condition in which two events do not influence each other's probabilities.

Simulation

A method used to model and analyze the probability of complex systems through repeated random sampling.

Variance of a Random Variable

$V(X) = E(X^2) - (E(X))^2$, representing the spread of a distribution.