Download Means and Variances of Random Variables

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 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
x = x1p1 + x2p2 + ...+ xkpk
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 (s2) and the variance of a
random variable we need to change our notation to x2. 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 - x)2 of the variable X
from its mean. See page 410 for more detail.
VARIANCE:  X2  ( x1   X )2 p1  ( x2   X ) 2 p2  ...  ( xk   X ) 2 pk
STANDARD DEVIATION =  X = SQUARE ROOT OF VARIANCE
The LAW OF LARGE NUMBERS (holds true for any population)
Draw independent observations at random from any population with finite mean 
Decide how accurately you would like to estimate  As the number of observations
drawn increases, the mean  x of the observed values eventually approaches the mean
of the population as closely as you specified and then stays that close. The law says
broadly that the average results of many independent observations are stable and
predictable and that averaging over many individuals produces a stable result.
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 mean
outcome is close to the distribution mean.
RULES FOR MEANS: (READ p 418-419)
RULES FOR VARIANCES: (READ p 420-423)
Any linear combination of independent random variables is also normally
distributed. (See example 7.14 on p 424)
Don’t forget: Z =
xx


x  mean
, the area under the normal curve = probability
SD