Random Variables and their Probability Distributions

Random Variable

(a.k.a. chance variable, a stochastic variable, or simply a variate) is a function whose domain is a sample space and whose range is some set of real numbers.

Discrete Random Variable

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

Continuous Random Variable

is 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)

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. This function may be viewed or can be represented as a table, graph, or formula

Probability Distribution

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

function

f from Set A to Set B is a relation in which each element of the domain is paired with exactly one element of the range.

domain

of a function is defined as the set of all possible input values (commonly the x variable), which produces a valid output (y-value) from a particular function. In simple language, this is what can go into a function.

range

is the set of all possible output values (commonly the variable y, or sometimes expressed as f(x)), which results from using a particular function. In simple language, this is what actually comes out of a function.

Expected value of a random variable

(a.k.a. mean of a probability distribution) is the summation of each value of the variable multiplied by its probability

Characteristics of the Binomial Distribution

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

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

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

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

trial

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

independent

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

two (2) mutually exclusive outcomes

For each trial, there are_____________, success or failure.

probability of success is fixed

The ________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

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

Poisson Distribution

is very useful in decision-making with respect to quality control situation, waiting line problems (queue), and other application to business.

Simeon Denis Poisson

He developed Poisson Distribution and a French mathematician

Characteristics of the Poisson Distribution

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

2. The average number of outcomes of interest per time or space interval is

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.

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

𝜇

is the mean number of occurrences per unit (time, volume, area, etc.)

𝑒

is a constant approximately equal to 2.71828... (Actually, 𝑒 is the base of the natural logarithm system.)

𝑥

= number of occurrences (0, 1, 2, …)