DISCRETE & CONTINUOUS RANDOM VARIABLES
~The concepts of a discrete and
continuous random variables are of up most importance in statistics
& are often not understood well. Here is a brief critique of these
basic concepts.
~When a procedure is performed, we have certain outcomes (events). We call the set of all possible outcomes, the Sample space.
~For example, if we list all possible gender sequences for 3 children
born, the sample space would be MMM, FMM, MFM, MMF, FFM, FMF, MFF, FFF.
~If we assign a numerical value to each outcome, then the set of these
values would constitute the random variable x. In this case, if we were
interested in the number of girls, we would assign 0 to MMM, 1 to each
of FMM, MFM, MMF, 2 to each of FFM, FMF,
MFF, and & 3 to FFF.
~In this case, the values for x are 0,1,2,3. Since this set is finite (definite #), it is said to be discrete.
~Any set of values for x that is finite is discrete.
~An infinite set that can be placed into a 1-1 correspondence with the
set of natural numbers, 1,2,3,4,5,6, & so on, is said to be
"countably infinite" and will also form a discrete set.
~Note that there are always "gaps" between values in a discrete set of numbers.
(i.e., in the above example, x values between the integers cannot be taken on)
~On the other hand, if our procedure deals with picking a point inside
a circle of radius 11, then our sample space would consist of all such
possible points.
~If we assign a value to each point which gives the distance that point
is away from the center of the circle, then these values would consist
of an infinite, uncountable, continuous range of values between 0 and
11 (that's how far away from the center each randomly selected point
would have to be). Every number between 0 and 11 can occur (no "gaps"
between numbers).
~In this case, the random variable x would be continuous.
~For the most part, in this course, we will deal with discrete random variables.
Continuous variable functions (standard normal, t-curves, & the
Chi-square distributions, among others) will be used to give reasonable
approximations to discrete distributions as long as certain conditions
hold. In a few cases, some minor adjustments must be made.
~Probabilities can then be tied in with the chance occurrences of each event.
~Important Note: For the continuous case, the probability that x takes
on one particular value will be zero (i.e., In a given population of
individuals, the probability that a person is selected with a weight of
exactly 123 lbs is zero). We use intervals instead of single values to
compute probabilities. These are directly related to the areas under
each probability density function over a given interval for x.
Probabilities based on the Normal curve (most important probability
density function in basic statistics) are found this way.