Basics of Statistics & Mathematics – Unit 4: Theoretical Distribution

Vocabulary flashcards covering essential terms from Unit 4 – Theoretical Distribution, including probability concepts, counting rules, and key discrete and continuous distributions.

Theoretical Distribution

A probability-based mathematical model that predicts how values are expected to behave under ideal conditions (e.g., normal, binomial, Poisson, exponential).

Empirical Distribution

A distribution derived from observed data rather than theoretical probability rules.

Random Experiment

A process of measurement or observation with uncertain outcome but well-defined possible results.

Outcome

A single possible result of a random experiment or trial.

Sample Space (S)

The set of all possible outcomes of a random experiment.

Event

Any subset of outcomes from the sample space; a ‘simple event’ cannot be decomposed further.

Mutually Exclusive Events

Events that cannot occur simultaneously in a single trial.

Collectively Exhaustive Events

A set of events that includes every possible outcome of the experiment.

Independent Events

Events whose occurrence does not affect each other’s probabilities.

Dependent Events

Events where the occurrence of one influences the probability of the other.

Compound Event

An event formed by the simultaneous occurrence of two or more simple events.

Equally Likely Events

Events that have the same probability of occurring.

Complementary Event (Ā)

All outcomes in the sample space that are not in event A.

Classical Approach (to Probability)

Probability defined as favourable outcomes divided by total equally likely outcomes.

Relative Frequency Approach

Probability estimated as the proportion of times an event occurs in repeated trials.

Subjective Approach

Probability assigned by personal judgment or belief when data are insufficient.

Fundamental Rules of Probability

Probabilities lie between 0 and 1; total probability of the sample space equals 1; P(ϕ)=0; P(Ā)=1−P(A).

Factorial (n!)

Product n × (n–1) × … × 1, with 0!=1; used in counting permutations/combinations.

Permutation (nPr)

An ordered arrangement of r items selected from n distinct items: n!/(n−r)!.

Combination (nCr)

An unordered selection of r items from n distinct items: n!/[r!(n−r)!].

Multiplication (Counting) Rule

If stage 1 has n₁ ways, stage 2 has n₂ ways, …, stage k has nₖ ways, total ways = n₁·n₂·…·nₖ.

Joint Probability

Probability that two (or more) events occur together, P(A ∩ B).

Conditional Probability

Probability of event A given event B has occurred, P(A|B).

Bayes’ Theorem

Formula that updates prior probabilities to posterior probabilities using new evidence: P(Ai|B)=P(B|Ai)P(Ai)/ΣP(B|Aj)P(Aj).

Prior Probability

Initial probability of an event before new information is considered.

Posterior Probability

Revised probability of an event after incorporating new evidence.

Random Variable

Numerical value assigned to each outcome of a random experiment.

Discrete Random Variable

Takes countable integer values (e.g., number of items sold).

Continuous Random Variable

Can assume any value within a range (e.g., time, distance).

Bernoulli Process

Repeated trials with only two mutually exclusive outcomes: success or failure.

Binomial Distribution

Discrete distribution giving the number of successes in n independent Bernoulli trials with constant success probability p.

Binomial Probability Function

P(X=r)=nCr pʳ qⁿ⁻ʳ, where q=1−p.

Mean of Binomial Distribution

μ = n p.

Standard Deviation of Binomial

σ = √(n p q).

Poisson Distribution

Discrete distribution describing the number of events occurring in a fixed interval when they happen at a constant average rate λ and independently.

Poisson Probability Function

P(X=r)=λʳ e^{−λ}/r!, r=0,1,2,…

Poisson–Binomial Approximation

Poisson approximates binomial when n is large and p is small (λ = np).

Mean/Variance of Poisson

Both equal λ (μ = σ² = λ).

Probability Density Function (PDF)

Function describing how probability is distributed over values of a continuous random variable; area under the curve equals 1.

Continuous Probability Distribution

Distribution defined by a PDF over an interval rather than individual points.

Normal Distribution

Symmetric bell-shaped continuous distribution defined by mean μ and standard deviation σ.

Standard Normal Distribution

Normal distribution with μ = 0 and σ = 1.

Z-Statistic (Z-Score)

Standardized value z = (x − μ)/σ indicating how many standard deviations x is from the mean.

Exponential Distribution

Continuous, non-symmetric distribution describing time or distance between Poisson events; PDF: f(x)=λe^{−λx}, x≥0.

Rate Parameter (λ)

Reciprocal of the mean in an exponential distribution; equals average number of events per unit time.

Mean/Std Dev of Exponential

Both equal 1/λ.