Q: How many
statisticians does it take to change a light bulb?

A: One—plus or minus three.

Chapter 7

Sec 7.2

Probability is the math language that describes the LONG-RUN
regular behavior of* random phenomena*.

Read the first sentence again until you understand every word.

The mean
`x of a
set of observations is their ordinary average. The mean of a random
variable X is also an average of the possible values of X, but with an essential
CHANGE to take into account the fact that NOT all outcomes need be equally
likely. See Ex.7.5 page 407. The mean of X is the LONG RUN AVERAGE
you expect for a very large number of times. Just as probabilities are an*
idealized* description of long run proportions, the __mean of a probability
distribution describes the long run average outcome__.

The common symbol for
the mean of a probability distribution is
m_{x
}...notice the subscript to indicate this is the mean of a random variable
X and not the mean of a normal distribution. The mean of a random variable
X is often called the EXPECTED VALUE of X. The mean of a discrete random
variable is the average of the possible outcomes, but a __weighted __average in
which each outcome is __weighted by its probability__. Because the
probabilities add to 1, we have total weight 1 to distribute among the outcomes.
The probability distribution of a discrete random variable is given in table
form as on page 408 with row 1 giving variable values and row 2 giving
corresponding probabilities. To find the mean of X, multiply each possible
value by it probability, then ADD. Symbolically, it looks like

m_{x} = x_{1}p_{1}
+ x_{2}p_{2 }+ ...+ x_{k}p_{k
}The mean is a measure of the center of a distribution. The variance
and the standard deviation are the measures of spread that accompany the choice
of the mean to measure center. To distinguish between the variance of a
data set (s^{2})_{ }and the variance of a random variable we
need to change our notation to s_{x}^{2}._{
}The definition of the variance of a random variable is similar to
the definition of the sample variance from Chapter 1. That is, the
variance is an average of the squared deviation (X -
m_{x})^{2} of the variable X
from its mean. See page 410 for more detail.

**The "LAW OF LARGE NUMBERS"**..(holds true for any population)

Draw independent observations at
random from any __population __with finite mean m.
Decide how accurately you would like to estimate m.
*As the number of observations drawn increases, the mean **
` x of the
observed values eventually approaches the mean **
m *__
of the population__ as closely as you specified and then stays that
close. (asymptotic - remember????) The law says broadly that t

The mean of a random variable is the average of the variable in two senses:

1) by definition it is the average of the possible values, weighted by their probabilities

2) by the law of large numbers it is the long run average of many independent observations on the variable.

We are unable to distinguish random behavior from systematic influences which points out the need for statistical inference to supplement exploratory analysis of data. Probability calculations can help verify that what we see in the data is more than a random pattern. How large is large depends on the variability of the random outcomes. The more variable the outcomes, the more trials are needed to ensure that the man outcome is close to the distribution mean.

**RULES FOR VARIANCES:**

The mean of a sum of random
variables is the sum of their means, BUT this addition is not always true for
variances. If random variables are independent the association between
their values is ruled out and their variances DO ADD. Two random variables
X and Y are independent if knowing that any event involving X alone did or did
not occur tells us nothing about the occurrence of any event involving Y alone.
Probability models often __assume independence__ when the random variables
describe outcomes that appear unrelated to each other. *You should ask
in each instance whether the assumption for independence seems reasonable.*

The exact rules for variance can be found on pages 420 and 421. See Combining normal random variables on page 424.