assumes:
Example 1
(X= sum of the numbers showing on two dice.) 
Example 2
Y = Payoff from simple dice game. roll a die:
if 1, 2, or 3 win $10 (+10) the experiment can result in six possible outcomes

A random variable must take on numerical values.
If the experiment was throw a die and win $10 on a 1 or 2 and lose $10 on a 4, 5, or 6, then the experiment would not be a random variable because it is not defined for all outcomes, namely "roll a 3".
A random variable is unknown before the experiment is carried out, but after the experiment is carried out the value of the random variable is always known.
Aside: Countable b/c some infinite sets can be used (for Poisson).
1/1 
2/1 
3/1 
4/1 
... 
1/2 
2/2 
3/2 
4/2 
... 
1/3 
2/3 
3/3 
4/3 
... 
1/4 
2/4 
3/4 
4/4 
... 
... 
... 
... 
... 

Example 3
A manufacturing plant produces a piece of metal with two holes whose specifications require the distance between the centers to be 3.000 ± 0.004 inches. This random variable can assume any value in between even though our ability to measure may require us to round off and work with discrete looking numbers. 
Example 4
A shipment contains 20 machines, 4 of which are defective. The firm receiving the shipment chooses a random sample of 3 machines (w/o replacement); if any of the machines in the sample are defective they reject the shipment.
solution

where the summation is over all values that X takes on. This is because these values of X are mutually exclusive and one of them must occur.
Example 5

Probability distributions can be represented by a table, graph, or a function.
Example 6
X = # of heads in two tosses of a fair coin.
table:
graph:
function:

The expected value of a random variable is analogous to the mean of a frequency
distribution.
(take class, multiply by the weight and divide by the sum of the weights,
which in probability is 1.)
so the expected value for the number showing on a true die:
Example 7
Suppose a trader is considering buying a bond issued by a financially troubled company. The price of the bond is $420. If the company avoids bankruptcy, the bond will be worth $1000. If the company declares bankruptcy, the bond will be worth nothing. He believes the probability of avoiding bankruptcy is 0.40 and that the probability of declaring bankruptcy is 0.60. so X = payoff from this investment
Should he accept this investment? (if he were to repeat this gamble again and again, would he tent to come out ahead or would he tend to lose?) The answer depends on E(X). so
so on average he would expect to lose $20 per gamble. If he found another bond with, say, an expected value of $50 he may decide to buy it rather than the one with E(X) = $20. 
(analogous to variance of a frequency distribution)
Variance:
and likewise
Standard Deviation:
The standard deviation of a random variable indicates the extent of the dispersion or variability among the values that the random variable may assume.