prob-stats e2

Random Variable (RV)

Describes the outcomes of a statistical (random) experiment in words.

Uppercase letters

Denote a random variable, such as X, Y. for example: The number of heads when flipping a coin 3 times.

Lowercase letters

Denote specific values of the random variable, such as x, y. for example: 2 means that in one trial, exactly 2 heads were observed.

Discrete Random Variables

Can assume only a countable number of values (finite or countably infinite).

Continuous Random Variables

Can assume measurable values.

Discrete Probability Distribution Function (PDF)

The list of all possible pairs (xi,P(xi)), representing the probabilities of discrete outcomes.

Conditions of Probability

Each probability must be between 0 and 1, and the sum of all probabilities must be 1.

Probability Distributions

Represented as tables, graphs, or functions that assign probabilities to outcomes.

Expected Value (Mean)

Represents the long-term average of the distribution.

Variance

Measures the spread of values in a probability distribution.

Standard Deviation

The square root of variance, measuring dispersion in data.

Binomial Distribution

A probability model with a fixed number of trials, two outcomes, and constant probability of success.

Fixed number of trials

Characteristic of binomial distribution denoting the number of attempts in the experiment.

Probability of success (p)

A constant probability assigned to each trial in a binomial distribution.

Independence of trials

In a binomial distribution, each trial is not influenced by previous trials.

Mean of Binomial Distribution (μ)

Calculated as μ = np.

Variance of Binomial Distribution (σ²)

Calculated as σ² = npq.

Standard Deviation of Binomial Distribution (σ)

Calculated as σ = √(npq).

Probability formula for Binomial Distribution

P(x) = (n choose x) p^x q^(n-x).

Geometric Distribution

Describes the number of Bernoulli trials until the first success.

Bernoulli Trials

Trials with two possible outcomes: success and failure.

Mean of Geometric Distribution

Calculated as μ = 1/p.

Variance of Geometric Distribution (σ²)

Calculated as σ² = (1 - p)/p².

Probability formula for Geometric Distribution

P(X = n) = q^(n-1) p.

Hypergeometric Distribution

Describes selection without replacement from a finite population.

Mean of Hypergeometric Distribution

Calculated as μ = n(r/N).

Variance of Hypergeometric Distribution (σ²)

Calculated as σ² = n(r/N)(N - r/N)(N - n/N - 1).

Poisson Distribution

Counts the number of occurrences of an event in a fixed interval.

Mean of Poisson Distribution (μ)

Equal to λ, representing the average rate of occurrence.

Variance of Poisson Distribution

Equal to λ, indicating the spread of the distribution.

Probability formula for Poisson Distribution

P(x) = (λ^x e^(-λ))/x!.

Continuous Probability Distributions

Describes continuous outcomes with probabilities as areas under curves.

Probability Density Function (pdf)

Graph representation of probabilities for continuous distributions.

Total area under the curve

In a continuous distribution, always sums to 1.

Uniform Distribution

All values within an interval are equally likely.

Mean of Uniform Distribution

Calculated as μ = (a + b)/2.

Variance of Uniform Distribution (σ²)

Calculated as σ² = (b - a)²/12.

Exponential Distribution

Measures the time between occurrences of an event.

Memoryless property

In exponential distribution, future probabilities are not affected by past events.

Probability Density Function for Exponential Distribution

f(x) = λe^(-λx).

Mean of Exponential Distribution (μ)

Calculated as μ = 1/λ.

Variance of Exponential Distribution (σ²)

Calculated as σ² = 1/λ².

Sampling without replacement

Choosing items from a population where previously chosen items are not returned.

Countable values

Values that can be listed or counted, characteristic of discrete random variables.

Independent events

In probability, events are independent if the occurrence of one does not affect the others.

Success in probability

The desired outcome in a set of trials, typically identified in distributions.