I've tried to be intuitive with these comments - rather like the first comments I give my students at the beginning of probability.
- Random variables: The ''variable'' part of the name indicates that you are studying something that isn't constant. The ''random'' part tells you that you are studying a variable whose values are governed by some type of probability rule. Here is an extremely simple example. Suppose we have a fair die - a die for which all six sides are as equal in area and weight as can be. Each side is marked with either 1, 2, 3, 4, 5 or 6 dots. An obvious random variable for this setting is the number of dots the side that is on top after a roll of the die. The number of dots will vary from roll to roll, and since it is impossible to precisely predict which side will come up, the side is random.
- What is P(X = x)? - You will typically see capital (upper-case) letters used as names for random variables. In my die example above I could say "Let X be the number of dots on the top side of the die after a single roll". Similarly, you will see lower-case letters in formulas used to indicate specific values of a random variable. Again, in my example, if I wanted to know the probability that after one roll the side with 4 dots would be up, I am asking for \Pr(X = 4). This leads to your next question.
- Probability distribution of a random variable? Every random variable has a probability distribution. Probability distributions tell us the rules for calculating probability for random variables. A distribution may be a table, or a formula, or be described by areas underneath curves (calculus is needed for the final situation).
General comments about probability distributions.
- Every probability distribution contains information that allows you to calculate any probability that relates to the associated random variable
- The total probability associated with a distribution is 1 (100%)
- Every probability you, or anyone, calculates, is between 0 and 1.
Think of my die example one final time. It seems reasonable to assume the sides all of the same chance of being on top when the die is rolled. The probability distribution can be written as a table this way:
<br />
\begin{tabular}{lcccccc}<br />
X & 1 & 2 & 3 & 4 & 5 & 6 \\<br />
p(x) & 1/6 & 1/6 & 1/6 & 1/6 & 1/6 & 1/6 \\<br />
\end{tabular}<br />
What is the probability that the side with three dots comes up? it is
<br />
\Pr(X = 3) = p(3) = \frac 1 6<br />
What is the probability the number of dots will be two or fewer?
<br />
\Pr(X \le 2) = p(1) + p(2) = \frac 1 6 + \frac 1 6 = \frac 2 6 = \frac 1 3<br />
Finally, it is possible to write this same distribution as a function. It is
<br />
p(x) = \frac x 6, \quad x = 1, 2, 3, 4, 5, 6<br />
The question of which is preferred depends on the people using it.
Hope this helps.
