Quantitative Prefi

Flashcards on Random Variables and Probability Distributions

Random Variable

A function whose domain is a sample space and whose range is some set of real numbers.

Discrete Random Variable

A random variable that may assume a finite or countable number of possible outcomes that can be listed.

Continuous Random Variable

A random variable that may assume an uncountable number of values or possible outcomes, represented by the intervals on a number line.

Probability Mass Function (pmf)

It provides the probabilities 𝑓(𝑥) = 𝑃(𝑋 = 𝑥) for all possible values that a discrete random variable (𝑥) can take on in the range of 𝑿.

Probability Distribution

A function that describes the shape, character, and relative likelihoods of obtaining the possible values that a random variable can assume.

Function

A relation in which each element of the domain is paired with exactly one element of the range.

Domain of a Function

The set of all possible input values (commonly the x variable), which produces a valid output (y-value) from a particular function.

Range of a Function

The set of all possible output values (commonly the variable y, or sometimes expressed as f(x)), which results from using a particular function.

Expected Value of a Random Variable

The summation of each value of the variable multiplied by its probability.

Mean of a Probability Distribution

The value that is expected to occur on average.

Variance of a Random Variable

Σ(𝑋 − 𝜇𝑥)2 ⋅ 𝑃(𝑋 = 𝑥)

Standard Deviation of a Random Variable

√𝛅𝟐

Characteristic of the Binomial Distribution (1)

The experiment is performed for a fixed number of times. Each repetition of the experiment is called a trial.

Characteristic of the Binomial Distribution (2)

The trials are independent. This means that the outcome of one (1) trial will not affect the outcome of the other trials.

Characteristic of the Binomial Distribution (3)

For each trial, there are two (2) mutually exclusive outcomes, success or failure.

Characteristic of the Binomial Distribution (4)

The probability of success is fixed for each trial of the experiment.

Binomial Distribution Formula

Where: • 𝑛 = the number of trials (sample size) • 𝑝 = the probability of a success on any single trial • 𝑟 = the number of successes in sample, (r = 0, 1, 2, …, n) • 𝑞 = 1 − 𝑝 = the probability of a failure

Characteristic of the Poisson Distribution (1)

The outcomes of interest are rare relative to the possible outcomes.

Characteristic of the Poisson Distribution (2)

The average number of outcomes of interest per time or space interval

Characteristic of the Poisson Distribution (3)

The number of outcomes of interest is random, and the occurrence of one (1) outcome does not influence the chances of another outcome of interest.

Characteristic of the Poisson Distribution (4)

The probability of an outcome of interest that occurs in a given segment is the same for all segments.

Poisson Distribution Formula

Where: • 𝜇 = is the mean number of occurrences per unit (time, volume, area, etc.) • 𝑒 = is a constant approximately equal to 2.71828… • 𝑥 = number of occurrences (0, 1, 2, …)