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Appendix A4: Normal Distribution


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Basic Definitions and Facts from Statistics A random variable can be viewed as the name of an experiment

with a probabilistic outcome. Its value is the outcome of the

experiment. A probability distribution for a random variable Yspecifies the

probability Pr(Y=yi) that Ywill take on the value of yi for eachpossible value of yi.

The expected value, or mean, of a random variable Yis .

The symbol is commonly used to represent E[Y].

The standard deviation of Yis .

The symbol is often used to represent the standarddeviation of Y.

)Pr(][ == i ii yYyYE

)(YVar

Y

Y


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Mean

Expected Value (also called mean value) is the average

of the values taken on by repeatedly sampling therandom variable

Definition: Consider a random variable Y that takes on

the possible values y1,yn. The expected value of Y,E[Y], is

If Y takes value 1 with prob .7 and value 2 with prob .3then expected value is 1x0.7+2x0.3=1.3

)Pr(][

1

i

n

i

i yYyYE ==


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Mean and Variance

Variance captures how far the random variable is expected to vary

from its mean value

Definition: The variance of a random variable Y, Var[Y], is

If Y is governed by a Binomial DistributionE[Y] = np

e.g., if n =50 and p = .2 then E[Y]= 25

]])[[(][ 2YEYEYVar


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Standard Deviation

The square root of the variance is called the standard

deviation

Definition: The standard deviation of a randomvariable Y is

]2

])[[( YEYEY


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Basic Definitions and Facts

The Binomial distribution gives the probability of

observing rheads in a series of n independent cointosses, if the probability of heads in a single toss isp.

The Normal distribution is a bell-shaped probabilitydistribution that covers many natural phenomena.

The Central Limit Theorem states that the sum of a

large number of independent, identically distributedrandom variables approximately follows a Normaldistribution.


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The Binomial Distribution

Probability of

observing r headsfrom n coin flip

experiments

Form of Binomial

distribution depends

on sample size n

and probability por errorD(h)


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Basic Definitions and Facts, continued

An estimator is a random variable Yused toestimate some parameterp of an underlyingpopulation.

The estimation bias of Y as an estimator for pis the quantity (E[Y]-p). An unbiasedestimator is one for which the bias is zero.

An N% confidence interval estimate forparameter p is an interval that includes p withprobability N%.

Table 5.2


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Estimators, Bias and Variance

Definition: The estimation bias of an estimator Y for an

arbitrary parameter p is

If the estimation bias is zero, then Y is an unbiasedestimator for p

errorS(h) is an unbiased estimate of errorD(h), sinceexpected value of r is np, and since n is constantexpected value of r/n is p

pYE

][


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The Normal Distribution

Table 5.4


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Normal or Gaussian Distribution

Well studied

Tables specify the size of the interval about the meanthat contains n% of the probability mass under thenormal distribution


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The Normal Distribution

A bell-shaped distribution defined by the probability density function

If the random variable x follows a normal distribution, then

The probability that X will fall into the interval (a,b) is given by

The expected, or mean, value of X, E[X], is

The variance of X, Var(X) is

The standard deviation of X, , is

2)(2

1

22

1)(

=x

exp

b

adxxp )(

=][XE

2)( =xVar

2

=x


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Normal Distribution, Mean 0, Standard Deviation 1

With 80% confidence the r.v. will lie in the two-sided interval[-1.28,1.28]


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Parameters of Normal Density Satisfy:


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Normally distributed samples tend to cluster

around the mean


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Standardized Random Variable

Mahanalobis distance, also z-score in one-dimension

Probability is 0.95 that Mahanalobis distance

from x to mu is les than 2


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Standardized r.v. has zero mean unit std

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