Variance and Standard Deviation - University of Pennsylvania
[Pages:49]Variance and Standard Deviation
Christopher Croke
University of Pennsylvania
Math 115 UPenn, Fall 2011
Christopher Croke
Calculus 115
Variance
The first first important number describing a probability distribution is the mean or expected value E (X ).
Christopher Croke
Calculus 115
Variance
The first first important number describing a probability distribution is the mean or expected value E (X ). The next one is the variance Var (X ) = 2(X ). The square root of the variance is called the Standard Deviation.
Christopher Croke
Calculus 115
Variance
The first first important number describing a probability distribution is the mean or expected value E (X ). The next one is the variance Var (X ) = 2(X ). The square root of the variance is called the Standard Deviation. If f (xi ) is the probability distribution function for a random variable with range {x1, x2, x3, ...} and mean ? = E (X ) then: Var (X ) = 2 = (x1 -?)2f (x1)+(x2 -?)2f (x2)+(x3 -?)2f (x3)+...
It is a description of how the distribution "spreads".
Christopher Croke
Calculus 115
Variance
The first first important number describing a probability distribution is the mean or expected value E (X ). The next one is the variance Var (X ) = 2(X ). The square root of the variance is called the Standard Deviation. If f (xi ) is the probability distribution function for a random variable with range {x1, x2, x3, ...} and mean ? = E (X ) then:
Var (X ) = 2 = (x1 -?)2f (x1)+(x2 -?)2f (x2)+(x3 -?)2f (x3)+...
It is a description of how the distribution "spreads". Note Var (X ) = E ((X - ?)2).
Christopher Croke
Calculus 115
Variance
The first first important number describing a probability distribution is the mean or expected value E (X ). The next one is the variance Var (X ) = 2(X ). The square root of the variance is called the Standard Deviation. If f (xi ) is the probability distribution function for a random variable with range {x1, x2, x3, ...} and mean ? = E (X ) then:
Var (X ) = 2 = (x1 -?)2f (x1)+(x2 -?)2f (x2)+(x3 -?)2f (x3)+...
It is a description of how the distribution "spreads". Note Var (X ) = E ((X - ?)2).
The standard deviation has the same units as X . (I.e. if X is measured in feet then so is .)
Christopher Croke
Calculus 115
Problem: Remember the game where players pick balls from an urn with 4 white and 2 red balls. The first player is paid $2 if he wins but the second player gets $3 if she wins. No one gets payed if 4 white balls are chosen.
Christopher Croke
Calculus 115
Problem: Remember the game where players pick balls from an
urn with 4 white and 2 red balls. The first player is paid $2 if he
wins but the second player gets $3 if she wins. No one gets payed
if 4 white balls are chosen.
We have seen that the payout and probabilities for the first player
are:
Payout Probability
2
8 15
0
1 15
-3
6 15
Christopher Croke
Calculus 115
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