Lesson1_RandomVariable&ProbabilityDistribution.

RANDOM VARIABLES & PROBABILITY DISTRIBUTIONS

​WHAT IS A VARIABLE?

A variable is any information, attribute, number, characteristic, or quantity that describes a thing, a person, place, event, or idea that can be measured or counted. It can be qualitative or quantitative, wherein the latter can either be discrete or continuous.

​WHAT IS A Discrete VARIABLE?

A discrete variable is a quantitative variable whose value can ONLY be attained through COUNTING. The possible values can be finite in number or COUNTABLY FINITE if the counting process has NO END.

A random variable is a variable whose value is dependent on the outcome of a well-defined random event or experiment (e.g., throwing a pair of dice or drawing a card from a standard deck)

​In an experiment, the outcome is said to be a discrete random variable if the experiment only has a countable or countably infinite number of outcomes. No other outcome exists between two consecutive outcomes.

​The set of all possible outcomes in an experiment is called sample space

​The following are some examples of discrete variables:

a. The number of students who are fully vaccinated.

b. The number of children in a family.​

​c. The number of COVID - 19 Patients

d. Number of Planets

​

​Notice that the number of students, children, patients, and planets are measured as whole-number units. There can be1, 2, 3, or more students or children, but we can not have 2.7 students or children.

​A continuous variable is a quantitative variable that can assume an INFINITELY many, UNCOUNTABLE number of REAL NUMBER values. The value given to observation can include values as small as the instrument of measurement allows.

​In an experiment, the outcome is said to be a continuous random variable if an outcome can take an UNCOUNTABLY INFINITE number of possible outcomes within a specified real number interval. Here, it is always possible to have an outcome between any two existing ones.

​The following are some examples of continuous variables:

a. The distance traveled by a bicyclist during practice.

b. The exact age of a person.

c. Height and Weight of an athlete ​

​The possible values of a random variable are values that are obtained from functions that assign areal number to each point of a sample space.

​We can assign numeric values to these outcomes as {1, 2, 3, 4, 5, 6, 7, 8}. Thus, there are 8 possible outcomes or eight elements in the sample space.

​If we are interested in the number of wins (which is now the random variable),

​Possible Outcomes	​Value of the Random Variable Y (Number of Wins)
​Steps	​Solution:
​1. Determine the sample space. Let W represent Win and L represent loss	​The sample space for this experiment is: S={WWW, WWL, WLW, LWW, WLL, LWL, LLW, LLL}
​	​
​	​

​2. Count the number of tails in each outcome in the sample space and assign this number to this outcome	​
​	​

​Possible Outcome	​Value of the Random Variable W (Number of wins)
​WWW	​3
WWL​	​2
WLW​	​2
LWW​	​2
LLL​	​0
LLW​​	​1
LWL​​	​​1
WLLL​	​1

​So, the possible values of the random variable W are 0,1,2,3.

​The probability distribution function is a function P(X) that shows the relative probability that eachoutcome of an experiment will happen.

​From the previous Illustration, we form the probability distribution function P, as follows.

​where X is the number of wins in three games and P(X) is the probability of winning X number of times in 3 games.

​Consider tossing a pair of UNBIASED coins.

a. Determine its sample space.

b. Assign possible values to the sample points.

c. Construct a probability distribution for getting a HEAD,

​CHECKYOURPROGRESS

​Consider tossing an unbiased coin three times

a. Determine its sample space.

b. Assign possible values to the sample points.

c. Construct a probability distribution for getting a TAIL