Definitions
Random Variable  A variable whose values are determined by chance. Probability distribution  Consists of the values a random variable can assume and their corresponding probabilities

Probability Distributions
 2 requirements
(1.) The sum of the probabilities of all the events in the sample space must equal 1. (2.) The probability of each event in the sample
space must be between

Mean, Variance, and Standard
Deviation:
From Chapter 3: Mean: m= (S X)/N Here this formula can’t be used. For example, how would you get the mean number of spots that would show up when you rolled a particular die? If you rolled it 100 times (N = 100) and added them up, you’d get an approximate answer, but to get an exact answer N would have to be infinite!

So how do we find the mean of a probability distribution?
To find the mean of any probability distribution, you
must multiply each possible outcome in the sample space by its corresponding
probability of occurring, then sum the products!
Formula for the Mean: The mean of a probability distribution is given by: m=
X1 ^{.}P(X1)
+ X2 ^{.}P(X2)
+ X3 ^{.}P(X3)
+....+Xn ^{.}P(Xn)
where X1 , etc.
are the outcomes and P(X1), etc.,
are their corresponding probabilities
s = SQUARE ROOT{S(X  m)^{2}/N} But again, to get an exact answer, N would have to be infinite. Therefore we introduce the following formulas:

Formula for the Variance:
s^{2} = S[(X  m)^{2.}P(X)] But this formula can be tedious, subtracting the mean from each entry, so an algebraically equivalent formula can be derived that is much simpler to calculate: s^{2}
= S[X^{2} ^{.}P(X)]  m^{2}
Formula for the Standard Deviation:

Expectation
The expected value of a discrete random variable of a probability distribution is the theoretical average of the variable (aka the mean!) E(X) = m = SX^{.}P(X) The expected value is really a measure of the average of the losses in a game of chance. Note:

The Binomial Distribution
Many probability problems have only two possible outcomes, or can be reduced to two. Flip a coin? (heads or tails)
Situations like these are referred to as binomial experiments, and will satisfy the following four requirements: 1. Each trial can have only two outcomes, or outcomes that can be reduced to two outcomes. These can be thought of as either success or failure . 2. There must be a fixed number of trials 3. The outcomes of each trial must be independent 4. The probability of
success must remain the same for each trial

A binomial experiment and its results yield the binomial
distribution. This leads us to the formula:
P(X)
= {n!/(nX)!X!}_{p}^{X}_{q}^{nX}
P(X) Probability of X successes
How does this formula work? If you look at the first part of the formula (the part in the { }), you may notice that this is exactly the same formula we used when doing combinations! This gives us the number of ways to get X successes from n trials. Now, however, we have to figure in the mathematical probabilities (p) for these X successes (remember requirement 4) and the mathematical probabilities for failures (nX). This yields the formula above!

Computing these probabilities can be pretty boring, but
thankfully tables have been compiled for selected values of n and p. (See
Table 2, Page 597)
Mean, Variance, and Standard Deviation for the Binomial Distribution The mean, variance, and standard deviation for the binomial
distribution can be computed quite easily from the number of trials (n),
the mathematical probability of success (p) and the mathematical probability
of failure (q):
Mean: m = np Variance: _{s}^{2} = npq Standard Deviation: s = SQUARE ROOT{npq} 
Multinomial Distribution
If each trial in an experiment has more than two outcomes, then we must use the multinomial distribution. This distribution is discussed in the Schaums Outlines: Statistics, Chapter 7 (see syllabus). The same general requirements apply here as did in the Binomial distribution (except for the restriction of two outcomes only): 1. There must be a fixed number of trials 2. The outcomes of each trial must be independent 3. The probability of success must remain the same for each trial Note: The Multinomial Distribution is a general
distribution (three or more outcomes possible); therefore the Binomial
Distribution is really just a special case of the multinomial distribution
when only two outcomes are possible.

The Poisson Distribution
This probability distribution is useful when n is large, p is small, and the independent variables occur over a period of time. It can also be used when a density of items is distributed over a given area or volume. The formula is given by: P(X,l) = _{e}^{l.}_{l}^{X}_{/X!} Again, the computation of this equation will drive nearly
anyone over the brink if done enough times, so our author has thoughtfully
included a set of tables for selected n and ?.

Homework:
Read Chapter 5, sections 5.1 through 5.4. Please go to Lesson 7. 