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Review of Probability
Important Topics
1 Random Variables and Probability Distributions
2 Expected Values, Mean, and Variance
3 Two Random Variables
4 The Normal, Chi-Squared, Fm , , and t Distributions
5 Random Sampling and the Sampling Distribution
6 Large-Sample Approximations to Sampling Distributions
Definitions





Outcomes: the mutually exclusive potential results of a random
process.
Probability: the proportion of the time that the outcome occurs.
Sample space: the set of all possible outcomes.
Event: A subset of the sample space.
Random variables: a random variable is a numerical summary of
a random outcome.
Probability distribution: discrete variable





List of all possible [x, p(x)] pairs
 x = value of random variable (outcome)
 p(x) = probability associated with value
Mutually exclusive (no overlap)
Collectively exhaustive (nothing left out)
0  p(x)  1 for all x
 p(x) = 1
Probability distribution: discrete variable


Probabilities of events.
Cumulative probability distribution.
Example: Bernoulli distribution.

Let G be the gender of the next new person you meet, where
G=0 indicates that the person is male and G=1 indicates that
she is female.

The outcomes of G and their probabilities are
G= 1 with probability p
= 0 with probability 1-p
Probability distribution: continuous variable
1.
Mathematical formula
2.
Shows all values, x, and frequencies, f(x)
•
f(x) Is Not Probability
3. Properties
Frequency
(Value, Frequency)
f(x)
 f (x)dx  1
All x (Area Under Curve)
f (x )  0, a x  b
a
b
Value
x

Probability density function (p.d.f.).
Probability distribution: Continuous Variable

Cumulative probability distribution.
Uniform Distribution
1. Equally likely outcomes
2. Probability density
function
1
f ( x) 
d c
f(x)
1
d c
3. Mean and Standard Deviation
cd

2
d c

12
c
a
b
d
x
Expected Values, Mean, and Variance
Expected value of a Bernoulli random variable
Expected value of a continuous random variable

Let f(Y) is the p.d.f of random variable Y , then the expected
value of Y is
Variance, Standard Deviation, and Moments
Variance of a Bernoulli random variable
The mean of the Bernoulli random variable G is G  p
, so its variance is
The standard deviation is
Moments


The expected value of Y r is called the r th moments of the
random variable Y .
That is the r th moment of Y is E(Y r ).
The mean of Y , E(Y), is also called the first moment of Y .
Mean and Variance of a Linear Function of a
Random Variable
Suppose X is a random variable with mean
and
Then the mean and variance of Y are
and the standard deviation of Y is
and variance
,
Two Random Variables
Joint and Marginal Distributions
 The joint probability distribution of two discrete random
variables, say X and Y , is the probability that the random
variables simultaneously take on certain values, say x and  .
 The joint probability distribution can be written as the function

The marginal probability distribution of a random variable Y
is just another name for its probability distribution.
E (Y )  0  (0.15  0.15)  1 (0.07  0.63)
Conditional distribution of Y given X=x is
Conditional expectation of Y given X=x is
E ( M )  0  (0.35  0.45)  1 (0.065  0.035)  2  (0.05  0.01)
 3  (0.025  0.005)  4  (0.01  0.00)
 0.35
E ( M )  E ( M | A  0) Pr( A  0)  E ( M | A  1) Pr( A  1)
 (0  0.70  1  0.13  2  0.10  3  0.05  4  0.02)  0.5
(0  0.90  1 0.07  2  0.02  3  0.01  4  0.00)  0.5
 0.35

The mean of Y is the weighted average of the conditional
expectation of Y given X, weighted by the probability
distribution of X.


Stated differently, the expectation of Y is the expectation of the
conditional expectation of Y given X, that is
where the inner expectation is computed using the conditional
distribution of Y given X and the outer expectation is computed
using the marginal distribution of X.
This is known as the law of iterated expectations.
Conditional variance

The variance of Y conditional on X is the variance of the
conditional distribution of Y given X.
Var ( M | A  0)  [0  0.56]2  0.70  [1  0.56]2  0.13 
[2  0.56]2  0.10  [3  0.56]2  0.05 
[4  0.56]2  0.02
Var ( M | A  1)  [0  0.14]2  0.90  [1  0.14]2  0.07 
[2  0.14]2  0.02  [3  0.14]2  0.01 
[4  0.14]2  0.00
Independence


Two random variable X and Y are independently distributed,
or independent, if knowing the value of one of the variables
provides no information about the other.
That is, X and Y are independent if for all values of x and  ,

State differently, X and Y are independent if

That is, the joint distribution of two independent random
variables is the product of their marginal distributions.
Covariance and Correlation
Covariance
One measure of the extent to which two random variables move
together is their covariance.
Correlation

The correlation is an alternative measure of dependence between
X and Y that solves the “unit” problem of covariance.

The random variables X and Y are said to be uncorrelated if
Corr(X, Y) = 0.
The correlation is always between -1 and 1.

The Mean and Variance of Sums of Random
Variables
Normal, Chi-Squared, Fm , , and t Distributions
The Normal Distribution
The probability density function of a normal distributed random
variable (the normal p.d.f.) is

where exp(x) is the exponential function of x.
The factor
ensures that
The normal distribution with mean μ and
variance σ2 is expressed as N(μ, σ2 ).
Black swan theory

The black swan theory or theory of black swan events is a
metaphor that describes an event that comes as a surprise, has a
major effect, and is often inappropriately rationalized after the
fact with the benefit of hindsight.
 The theory was developed by Nassim Nicholas Taleb
 The disproportionate role of high-profile, hard-to-predict,
and rare events.
 The non-computability of the probability of the
consequential rare events using scientific methods (owing to
the very nature of small probabilities).
 The psychological biases that make people individually and
collectively blind to uncertainty and unaware of the massive
role of the rare event in historical affairs.


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The main idea in Taleb's book is not to attempt to predict black
swan events, but to build robustness against negative ones that
occur and be able to exploit positive ones.
Taleb contends that banks and trading firms are very vulnerable
to hazardous black swan events and are exposed to unpredictable
losses.
Taleb is highly critical of the widespread use of the normal
distribution model as the basis for calculating risk.


The standard normal distribution is the normal distribution
with mean μ= 0 and variance σ2 = 1 and is denoted N(0, 1).
The standard normal distribution is often denoted by Z and its
cumulative distribution function is denoted by Ф. Accordingly,
Pr(Z ≤ c)= Ф(c), where c is a constant.
Key Concept 2.4
Copyright © 2003 by Pearson
Education, Inc.
2-42
The Empirical Rule (normal distribution)
36 of 42
Copyright © 2011 Pearson Education, Inc.



90%: +- 1.69
95%: +- 1.96
99%: +- 2.58
The bivariate normal distribution

The bivariate normal p.d.f. for the two random variables X and
Y is
where
is the correlation between X and Y .
Important properties for normal distribution.
1. If X and Y have a bivariate normal distribution with covariance ,
and if a and b are two constants, then
2. The marginal distribution of each of the two variables is normal.
This follows by setting a = 1; b = 0 in 1.
3. If
= 0, then X and Y are independent.
The Chi-squared distribution



The Chi-squared distribution is the distribution of the sum of
m squared independent standard normal random variables.
The distribution depends on m, which is called the degrees of
freedom of the chi-squared distribution.
A chi-squared distribution with m degrees of freedom is denoted
.
Fm , distribution




where
and
are independent.
When n is ∞,
.
The Fm , distribution is the distribution of a random variable
with a chi-squared distribution with m degrees of freedom,
divided by m.
Equivalently, the Fm , distribution is the distribution of the
average of m squared standard normal random variables.
The Student t Distribution

The Student t distribution with m degrees of freedom is
defined to be the distribution of the ratio of a standard normal
random variable, divided by the square root of an independently
distributed chi-squared random variable with m degrees of
freedom divided by m.

That is, let Z be a standard normal random variable, let W be a
random variable with a chi-squared distribution with m degrees
of freedom, and let Z and W be independently distributed. Then

When m is 30 or more, the Student t distribution is well
approximated by the standard normal distribution, and t∞
distribution equals the standard normal distribution Z.
Random Sampling


Simple random sampling is the simplest sampling scheme in
which n objects are selected at random from a population and each
member of the population is equally likely to be included in the
sample.
Since the members of the population included in the sample are
selected at random, the values of the observations Y1, … , Yn are
themselves random.
i.i.d. draws.


Because Y1, … , Yn are randomly drawn from the same
population, the marginal distribution of Yi is the same for eacn i
=1, … , n. Y1, … , Yn are said to be identically distributed.
When Y1, … , Yn are drawn from the same distribution and are
indepentently distributed, they are said to be independently and
identically distributed, or i.i.d.
Sampling Distribution of the Sample Average
The sample average of the n observations Y1, … , Yn is

Because Y1, … , Yn are random, their average is random and has
a probability distribution. The distribution of is called the
sampling distribution of .
Mean and Variance of
Suppose Y1, … , Yn are i.i.d. and let
and variance of Yi . Then
and
denote the mean

Sampling distribution of Y if Y is normally distributed

The linear combination of normally distributed random variable
is also normally distributed (equation 2. 42)
E (Y )   y , var(Y )   y

Therefore,
Y
N ( y ,
 y2
n
)
Large-Sample Approximations to Sampling
Distributions
Two approaches to characterizing sample distributions.
 Exact distribution, or finite sample distribution when the
distribution of Y is known.
 Asymptotic distribution: large-sample approximation to the
sampling distribution.
Law of Large Numbers



The law of large numbers states that, under general conditions,
will be near
with very high probability when n is large.
The property that is near
with increasing probability as n
increases is called convergence in probability, or consistency.
The law of large numbers states that, under certain conditions,
converges in probability to
, or, is consistent for
.
Key Concept 2.6
Copyright © 2003 by Pearson
Education, Inc.
2-59
The conditions for the law of large numbers are
 Yi , i=1, …, n, are i.i.d.
 The variance of Yi ,
, is finite.
Developing
Sampling Distributions
Suppose There’s a Population ...
• Population size, N = 4
• Random variable, x
• Values of x: 1, 2, 3, 4
• Uniform distribution
© 1984-1994 T/Maker Co.
Population Characteristics
Summary Measures
Population Distribution
N

X
i 1
N
i
 2.5
N

 X
i 1
.3
.2
.1
.0
i
N
 
P(x)
x
1
2
 1.12
2
3
4
All Possible Samples
of Size n = 2
16 Samples
16 Sample Means
1st 2nd Observation
Obs 1
2
3
4
1st 2nd Observation
Obs 1
2
3
4
1
1,1 1,2 1,3 1,4
1 1.0 1.5 2.0 2.5
2
2,1 2,2 2,3 2,4
2 1.5 2.0 2.5 3.0
3
3,1 3,2 3,3 3,4
3 2.0 2.5 3.0 3.5
4
4,1 4,2 4,3 4,4
4 2.5 3.0 3.5 4.0
Sample with replacement
Sampling Distribution
of All Sample Means
16 Sample Means
1st 2nd Observation
Obs 1
2
3
4
1 1.0 1.5 2.0 2.5
2 1.5 2.0 2.5 3.0
3 2.0 2.5 3.0 3.5
4 2.5 3.0 3.5 4.0
Sampling Distribution
of the Sample Mean
P(x)
.3
.2
.1
.0
x
1.0 1.5 2.0 2.5 3.0 3.5 4.0
Comparison
Population
.3
.2
.1
.0
Sampling Distribution
P(x)
x
1
2
  2.5
  1.12
3
4
P(x)
.3
.2
.1
.0
x
1.0 1.5 2.0 2.5 3.0 3.5 4.0
 x  2.5
 x  .79
Chebyshev Inequality

For any random variable X ~ (μ,  ), and for any k > 0,

The inequality can be rewritten as

Therefore, the inequality says that at least with 1 −
probability, the random variable will fall within the region μ ± kσ
Proof of Weak LLN

For any X

According to Chebyshev Inequality, for any ε > 0, we have
Formal definitions of consistency and law of large
numbers.
Consistency and convergence in probability.
 Let S1 , S2 , … , Sn , … be a sequence of random variables. For
example, Sn could be the sample average of a sample of n
observations of the random variable Y .
 The sequence of random variables {Sn} is said to converge in
probability to a limit, μ, if the probability that Sn is within ±δ of
μ tends to one as n → ∞, as long as the constant is positive.

That is,
if and only if Pr [| Sn – μ |≥δ] → 0 as n → ∞ for every


δ > 0.
If
, then Sn is said to be a consistent estimator of μ .
The law of large numbers.
If Y1, … , Yn are i.i.d., E(Yi) =
and Var(Yi) < ∞, then
The Central Limit Theorem



The central limit theorem says that, under general conditions,
the distribution of is well approximated by a normal
distribution when n is large.
Since the mean of is
and its variance if
, when n is
large the distribution of is approximately N( , ).
Accordingly,
is well approximated by the standard normal
distribution N(0,1)
Convergence in distribution.


Let F1, … , Fn , … be a sequence of cumulative distribution
functions corresponding to a sequence of random variables, S1,
… , Sn , … .
Then the sequence of random variables Sn is said to converge in
distribution to S (denoted as
) if the distribution
functions {Fn} converge to F.

That is,
if and only if

where the limit holds at all
points t at which the limiting distribution F is continuous.
The distribution F is called the asymptotic distribution of Sn .
The central limit theorem
If Y1, … , Yn are i.i.d. and 0 <
< ∞, then
In other words, the asymptotic distribution of
is N(0,1) .
Slutsky’s theorem


Slutsky’s theorem combines consistency and convergence in
distribution.
Suppose that
, where a is a constant, and
. Then
Continuous mapping theorem
If g is a continuous function, then


But, how large of n is “large enough?”



The answer is: it depends on the distribution of the underlying Yi
that make up the average.
At one extreme, if the Yi are themselves normally distributed,
then is exactly normally distributed for all n.
In contrast, when Yi is far from normally distributed, then this
approximation can require n = 30 or even more.
Example: A skewed distribution.