Transcript Rules for Means of Random Variables

 A phenomenon is random if individual outcomes are
uncertain but there is nonetheless a regular
distribution of outcomes in a large number of
repetitions.
 Randomness requires a long series of independent
trials.
 The probability of any outcome of a random
phenomenon is the proportion of times the outcome
would occur in a very long series of independent
repetitions. That is, probability is a long-term relative
frequency of independent repetitions.
 Thus, the chance of something gives
the percentage of time it is expected
to happen, when the basic process is
done over & over again, independently
& under the same conditions.
A probability model is based on
independent trials. Besides
independent trials, it requires:
 A list of possible outcomes of the random
phenomenon: the sample space.
 A probability for each outcome or set of
outcomes—each event—of the random
phenomenon.
 Think of the probability model
as a box model.
See Freedman et al., Statistics.

E.g., someone has a set of 10 cards
consisting of three cards with #1, one card
with #2, two cards with #4, two cards with
#7, one card with #8, & one card with #10.
The cards are put in a box & mixed well.
 We make 5 draws from the box. After
each draw, the drawn card is replaced & the
cards are remixed. For each draw, what are
the chances of drawing the following
numbers: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10?
Card #
Probability (for
each card drawn)
1
.30
2
.10
4
.20
7
.20
8
.10
10
.10
Sum of event probabilities=1.0
 How did we compute the
probabilities for the box model?
We did so:
 By making a list of the possible outcomes
of the random phenomenon: the sample
space; &
 By assigning a probability for each
outcome (or set of outcomes) of the
random phenomenon—for each event.
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Card Number
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10
 Create the variables
. input cardnum prob
1. 1 .3
2. 2 .1
3. 4 .2
4. 7 .2
5. 8 .1
6. 10 .1
7. end
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. gr bar prob, over(cardnum)
1
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Card Number
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 To be valid, the assigned
probabilities must be premised
on independent, random
draws (i.e. independent,
randomly drawn observations).
 In the previous example about card
values, consider the set of cards as a
population.
 What we did, then, was draw a sample
from that population.
 What was the basis of the mean of the
sample drawn?
 It was the sample’s particular set
of draws from the population’s
sample space of events (i.e.
card values) & the probability
attached to each event.
How to do a probability histogram
(which in Stata can done via the ‘bar’
graph command)?
. gr bar prob, over(cardnum)

Histogram: breaks a quantitative continuous or discrete
variable’s range of values into intervals & displays the count
or percent of observations falling into each interval; density
(continuous) or frequency (discrete) histograms.
 Bar graph: displays the counts or percentages of the
components of a categorical variable.
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. gr bar prob, over(cardnum)
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Card Number
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Probability Rules
(1) The probability of an event A is between 0
& 1. That is, chances are between 0% &
100%.
(2) The summed probability of all events in a
sample space=1. That is, P(S) = 1.
(3)
The complement of any event A is
the event that A does not occur (e.g.,
live vs. die). Then P(Ac) = 1 - P(A).
That is, the chance of something
equals 100% minus the chance of the
opposite thing.
(4) Two events are disjoint (i.e. mutually
exclusive) if they have no outcomes in
common & thus can never occur
simultaneously (e.g., the probability of
being single or married). Then P(A or B)
= P(A) + P(B).
(5) If events A & B are independent, then
P(A and B) = P(A)P(B).
E.g., P(divorced and male).
Why is independence a tricky issue?
Remember that in the previous
chapter we began moving from
descriptive statistics to inferential
statistics.

 We’re about to examine methods of
inferential statistics in greater depth,
based on probability principles.
 The following, then, is about using
one or more samples to obtain
statistics, which in turn we’ll use to
estimate parameters.
 In particular, we’ll obtain sample
means & sample standard deviations,
which in turn we’ll use to estimate a
population’s mean.
We’ll begin to apply probability
principles toward this objective by
defining the following:
 Random variable
 Expected value of a random variable
 We’ll also do so by introducing a
modification to the standard deviation,
which will become the standard error.
 This modification will divide a
sample’s standard deviation by the
square root of the sample size-n.

With this modification, what varies is the
mean of samples from the population
mean. How is the meaning of standard
error, then, different from that of standard
deviation?
 Understanding this difference, as we’ll
see, is fundamental to understanding the
difference between descriptive &
inferential statistics.
 And again, what makes all of
this possible?
 The principles of probability.
Random Variable
 A random variable is a variable
whose value is a numerical outcome
of a random phenomenon.
 A discrete random variable X has a
finite number of possible values. The
probability distribution of X lists the
values & their probabilities.
 E.g., number of suicides in Miami-Dade
last year; number of votes by candidate in
Mexico’s presidential election; number of
members per household in South Africa;
number of housing types in Miami-Dade;
number of persons by gender in FIU.
 A continuous random variable X
takes values in an interval of numbers.
Its probability distribution is described
by a density curve. The probability of any
even is the area under the density curve
& above the values of X that make up the
event.
 E.g., age, weight, cholesterol level,
temperature; the proportion of adults
that opposes abortion rights for women;
the proportion of immigrants that sends
remittances to home countries.
 Because any density curve describes
an assignment of probabilities,
density distributions are
probability distributions.
Means, Variances & Standard
Deviations of Discrete Random
Variables
The mean of a random value X is called
the expected value of X: a weighted,
long-term average in which each
outcome is weighted by its probability.

 Expected value of a
discrete random variable:
x  x1 p1  x 2 p2  ...  xkpk  xipi
 The previous card example.
 Votes per candidate.
 Number of persons per household.
 Let’s calculate the expected value for
the previously discussed playing cards
example:
Card Number
1
2
4
7
8
10
Probability (for each draw)
.3
.1
.2
.2
.1
.1
 Each card value times its probability,
summing the results.
 Variance of a discrete random
variable: (each value – the
mean)squared, summing the results
 2  ( x1  x )2 p1  ...  ( xk  x )2 pk  ( xk  x )2 pk
Standard deviation of a discrete
random variable: square root of the
variance.

 Why do we refer to the expected
value of a random variable instead
of its mean?
Because we’re moving from
descriptive statistics to inferential
statistics.

 That is, from merely describing one
or more samples to using them to
make inferences about a population.
 In this regard, expected value
indicates that the mean of a random
variable is a long-term outcome of its
weighted probabilities.
 It is the value that we’d expect to
obtain if we could measure an entire
population.
 When we draw a sample, what
determines the sample mean? How
does this pertain to the expected
value?
The purpose of using the
expected value of a random
variable—the outcome of its
weighted probabilities over the long
term—becomes clearer when we
turn our attention to the Law of
Large Numbers.

Statistical Estimation & the Law of
Large Numbers
We’ve seen how sample means vary
from sample to sample.

 If the sample mean of a statistic rarely
captures the parameter & varies from
sample to sample, why is it nonetheless
a reasonable estimate of the parameter?
Because when based on a random
sample of independent observations:
 The sample mean is an unbiased
estimator of the population mean; &
 We can reduce its variability by
increasing the sample size.
 Another reason: The Law of Large
Numbers.
 If we keep adding observations to our
random sample of independent observations
(i.e. if we keep making the sample size-n
larger), the sample mean of a statistic is
guaranteed to get as close as we wish to the
parameter & to stay that close.
 That is, if we make a sample-n large
enough, eventually its mean outcome
gets close to the population mean &
stays close.
 This holds for any population, not
just for some special class such as
normal distributions.
 Based on the Law of Large
Numbers, the mean of a large
number of independent
observations is stable &
predictable.
The Law of Large Numbers versus the
‘Law of Small Numbers’:
 The Law of Large Numbers: probability
operates over the very long run.
 The ‘Law of Small Numbers’: in our daily
lives we tend to behave as if probability
operated in the short run.
Means & Variances of a
Random Variable

Rules for means of a random variable
 Rules for variances of a random
variable (& take the square root to
compute standard deviation).
Rules for Means of Random
Variables
 If X & Y are random variables, then the
mean of X + Y = mean of X + mean of Y
. mean of 7 dents per car + mean of 12
scratches per car = mean of 19 imperfections
per car
. mean household size of Dominicans in Miami
+ mean household size of Jamaicans in Miami
= mean D + mean J
 And if X is a random variable,
then multiplying the mean of X by
a constant = multiplying each
randomly selected observation by
a constant.
E.g., 3.23*mean income =
3.23*each randomly selected
income observation
 The former is a new random
variable that is based on a linear
transformation of the latter.
Addition Rule for Variances
If X & Y are independent random variables,
then:
 variance X+Y=variance X + variance Y
 variance X – Y = variance X + variance Y
Why? Because combining two variables in
any way adds their variation together.
X & Y are independent if rXY = 0
 E.g., the mean income of
Salvadorans in Miami + the mean
income of Haitians in Miami.
 The variance, & standard deviation,
of their combined incomes.
 Key question: Are the Salvadoran &
Haitian incomes independent of each
other? If not, see Rule 3 for
variances (page 330).
Let’s not lose sight of what we’re doing:
 By using both the expected value of the
sample mean of a random variable & its
standard deviation divided by the square root
of sample size-n (i.e. the standard error):
we’re moving from descriptive statistics to
inferential statistics.
 That is, instead of just describing one or
more samples, we’re now using the random
samples of size-n to make inferences about a
population.
What makes such inference possible?
The principles of probability, including:
 The fact that the sample mean is an
unbiased estimator of the population
mean & can be made less variable by
increasing sample size-n;
 & The Law of Large Numbers.
The rules of probability assure us
that one or more random samples
of independent observations of
size-n will give us sample means
that are reasonably accurate
estimates of a parameter.

What kind of statistics should be used
if a study involves not a sample but
rather a population?
 Descriptive statistics, not inferential
statistics, should be used in this instance.
 The reason is that there are no
sampling-based uncertainties involved
when analyzing a population.
Before analyzing a statistic, always ask:
 Is the statistic drawn from a random
sample of independent observations?
 If this premise is not reasonably
met, then the use of inferential
statistics is invalid.
Summary

What does random mean?
 What does probability mean?
 What does a probability model
require?
What are the rules of probability?
 What is a random variable? What
are its two basic types?
What’s an expected value of a
random variable, & how is it
computed?

 How are the variance & standard
deviation of the expected value
computed?
 Why do we use the expected value
of a random variable?
Why is the sample mean of a
statistic a reasonable estimate of a
parameter?

 What is the Law of Large
Numbers? What is the ‘Law of
Small Numbers’?
 What are the rules for means of a random
variable?
 What are the rules for variances of a
random variable? How do they pertain to
the standard deviation of a random
variable?
 If we have measures for a population or if
we don’t have a random sample, can we use
inferential statistics?