**This essay explains what various research terms mean, and provides examples for each.**

A **random variable** is a set of possible numerical outcome or value from a random experiment that is selected by chance. The outcome is a function from a sample space that is mapped into real numbers. It can be classified into: continuous random variable, which takes an infinite number of possible outcomes, or discrete which is countable.

*Example:*

Continuous: Experimenter wish to estimate the height of NBA basketball players. Measuring the height of each of the player rather inefficient, hence experimenter choose 30 players randomly from a set of all NBA players (sample space). Each basketball player in NBA has an equal chance of being selected. Thus each player selected for the sample is said to be a random variable. And height measurement can be anywhere between zero to infinity, hence continuous.

Discrete: number of heads when tossing a coin twice. Possible outcome or the sample space are HH, HT,TH,TT. Translated into the real line the possible number of heads, are 0,1 and 2.

**Random sample** is a subset of units or subjects from a population, where each unit in the population has an equal chance of being selected.

*Example:*

In a company consists of 300 employees (population), 30 people are selected randomly. The 30 people selected are called a random sample because they are selected by chance, each employee has an equal chance of being selected.

**Mgf** stands for moment generation function of x is denoted by M_x (t), exists if there is a positive number h such that the above summation exists and is finite for ‘h< t < h. By definition we can write mgf of X as:

Let X be a random variable with probability mass function f(x) and an element of S. Then:

M_x (t)=E(e^tx )=’_(x’S)”e^tx P(X=x)’ , if x is discrete

M_x (t)=E(e^tx )=’_(-‘)^”’e^tx f(x) dx’ , if x is continuous

Also, by theorem if X has mgf M_x (t) then

EX^n=’M_x’^((n) ) (0),

Where

‘M_x’^((n) ) (0)=d^n/’dt’^n M_x (t)|_0

Is the nth moment which can be found by deriving M_x(t) nth times and evaluating it at t=0.

Example:

Let x be distributed exponentially, hence the probably distribution function of x is

f(x)=1/?? e^((-x)/??) 0<x<‘ , ??>0

Obtaining the mgf we take:

E[e^tx ]=’_0^”'(e^tx 1/?? e^((-x)/??) [email protected] )

=1/(1-??t), t<1/??

Is the moment generating function of exponential distribution.

**Observational study** is a study where the investigators observe subjects and measure variables of interest without applying any treatments to the subjects.

*Example:*

A research study comparing the risk of developing heart disease between people who exercise regularly (at least three times a week for an hour each session) and people who do not exercise regularly. Experimenter only observe without altering any variable in the experiment or the environment.

**Experimental study** is a study where the investigators apply treatments to the experimental units and then measure the effect of the treatments on the variable of interest.

*Example:*

A research study comparing weight loss …

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