concept of a random variable

a random variable is the capital letter X

X then is a random variable

x (small x) denotes one of the values of the random variable X (big X)

example

two balls are drawn in succession without replacement from an urn containing 4 red balls and 3 black balls. The possible outcomes and the values y of the random variable Y is the number of red balls, are:

(THE right column is y)

discrete sample space: if a sample space contains a finite number of possibilities or an unending sequence with as many elements as there are whole numbers it is called a discrete sample space
e.g.: rolling a die until a 5 occurs.

(the latter half of that definition applies there)

On the other hand, there is the question "how far can a car drive on 5 liters of gasoline?"
the answer to which there are infinite possibilities. Therefore, for this latter, we say that it is a continuous sample space.

continuous sample space: if a sample space contains an infiinte number of possibilities equal to the number of points of a line segment, it is called a continuous sample space.

discrete random variable: represent count data
continuous random variable: usually are measured data

discrete probability distributions and discrete probability functions

tossing a coin 3 times:

sample space = hhh, hht, hth, thh, htt, tht, tth, ttt = 8 samples
if we want the probability of getting 2 heads and one tail == hht, hth, thh =
3/8

frequently it is convenient to represent all probabilities of a random
variable X by a formula.

for example we would write f(x) = P ( X = x)

that is, f(3) = P ( X = 3)

probability function == probability mass function == probability distribution

the set of ordered pairs (x, f(x)) is a {probability function , probability
mass function , probability distribution} if for each possible outcome x
1. f(x) >= 0
2. sum of f(x) = 1
3. P(X = x) = f(x)

I need to study how to answer: "A shipment of 8 similar microcomputers to a
retail outlet contains 3 that are defective. If a school makes a random
purchase of 2 of these computers, find the porobability distribution for the
number of defectives."

To solve this, please review counting.txt and cheat_sheet.txt in this
directory: you have to figure out the possibilitieis of choosing 2 from 8, of
choosing 1 from 3, of choosing 0 from 3, of choosing 2 from three, and also
then of choosing from the nondefective 2 from 5, 1 from 5 and 0 from 5
respectively.

I guess (tortuously) they are using "If an experiment can result in any one of
N different equally outcomes, and if exactly "n" of these outcomes correspond
to event "A", then the probability of event "A" is
P(A) = n/N

but here n = (combinations of defective)*(combinations of non-defective)
so P(A) = n/(total combinations of 2 out of 8)

do not confuse the above with CUMULATIVE DISTRIBUTION.

Now, these discrete distributions can be drawn with a "probability histogram"
or a "discrete cumulative" (step-ladder effect graph) or with a bar chart.
The x-axis is the values of x and the y-axis is their probability as dictated
by the function.

continuous probability distributions and continuous probability functions

density function
probability density function
cumulative distribution

joint probability distributions

joint probability distribution
probability mass function
joint density function
marginal distributions
conditional distributions

statistical independence