Essay: Aspects and features of research

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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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