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Statistical Review

This page contains the basic Rules for the Mean, Variance, Covariance, and Correlation for the expectation of random variables.  This summary can be extremely helpful if you do not work regularly in statistics or are a new student.  The proofs of these rules can be purchased for a nominal fee from the Order page.

The proofs are usually required problems or test questions in a non calculus first course in statistics.  The Rules and their proofs are part of a Statistical Review which is approximately 27 pages in 10 point type.  It is a handy review for someone who has been away from statistics for a while but suddenly finds an article using these Rules.  Students will find them helpful as well.  If you have MathType, you may edit the file.

FORMULAS AND RULES FOR EXPECTATIONS OF RANDOM VARIABLES

Formulas and Rules for the Mean of the Random Variable X

Formulas for the Mean

formulas for mean.gif (819 bytes)

where pi is the probability of the occurrence of  the value of xi.

Rules for the Mean

Rule 1.

The expectation of a constant, c, is the constant.
E(c) = c

Rule 2.

Adding a constant value, c, to each term increases the mean, or expected value, by the constant.
E(X+c) = E(X)+c

Rule 3.

Multiplying a random variable by a constant value, c, multiplies the expected value or mean by that constant.
E(cX ) = cE(X)

Rule 4.

The expected value or mean of the sum of two random variables is the sum of the means.   This is also known as the additive law of expectation.
E(X+Y) = E(X)+E(Y)

Formulas and Rules for the Variance, Covariance and Standard Deviation of Random Variables

Formulas for the Variance

variance formula 1.gif (620 bytes) or
variance formula 2.gif (497 bytes)   or
variance formula 3.gif (502 bytes)

Formulas for the Standard Deviation

standard deviation.gif (502 bytes)

Formulas for the Covariance

covariance fomula 1.gif (730 bytes) or
covariance formula 2.gif (576 bytes) or
Covariance formula 3.gif (519 bytes)

Rules for the Variance

Rule 1.

The variance of a constant is zero.
variance of c.gif (475 bytes)

Rule 2.

Adding a constant value, c, to a random variable does not change the variance, because the expectation (mean) increases by the same amount.
variance of X+c.gif (728 bytes)

Rule 3.

Multiplying a random variable by a constant increases the variance by the square of the constant.
variance of cX.gif (435 bytes)

Rule 4.

The variance of the sum of two or more random variables is equal to the sum of each of their variances only when the random variables are independent.
variance of X+Y.gif (594 bytes)

and in terms of the sigma notation
varsigmaxplusy.gif (355 bytes)
When two random variables are independent, sigxy0.gif (161 bytes) so that
variance when X,Y independent.gif (275 bytes)

Rules for the Covariance

Rule 1.

The covariance of two constants, c and k, is zero.
covariance of two constants.gif (634 bytes)

Rule 2.

The covariance of two independent random variables is zero.
covariance of independent X,Y.gif (248 bytes)

Rule 3.

The covariance is a combinative as is obvious from the definition.
covxyyx.gif (343 bytes)

Rule 4.

The covariance of a random variable with a constant is zero.
covariance of X,constant.gif (246 bytes)

Rule 5.

Adding a constant to either or both random variables does not change their covariances.
covcXkY.gif (421 bytes)

Rule 6.

Multiplying a random variable by a constant multiplies the covariance by that constant.
cxkycov.gif (422 bytes)

Rule 7.

The additive law of covariance holds that the covariance of a random variable with a sum of random variables is just the sum of the covariances with each of the random variables.
covariance x+y,z.gif (523 bytes)

Rule 8.

The covariance of a variable with itself is the variance of the random variable.   By definition,
cov(X,,X)=var(X).gif (847 bytes)

Formulas and Rules for the Correlation Coefficient of Random Variables

Rules for the Correlation Coefficient

Rule 1.

Adding a constant to a random variable does not change their correlation coefficient.
correlation coefficient.gif (1267 bytes)

Rule 2.

Multiplying a random variable by a constant does not change their correlation coefficient.  For two random variables
Z = a+bX and W = c+dY, where a,b,c, and d are constants,

correlation coefficient of linear function.gif (2408 bytes)

Because the square root of the variance is always positive, the correlation coefficient can be negative only when the covariance is negative.  This leads to

Rule 3.

The correlation coefficient is always at least -1 and no more than +1.
correlation coefficient limits.gif (203 bytes)

Formulas and Rules for the Sample Mean, Variance, Covariance and Standard Deviation, and Correlation Coefficient of Random Variables

Rules for Sampling Statistics

Rule 1.

The sample mean, xbar.gif (102 bytes) is computed by
sample mean.gif (455 bytes)

Rule 2.

The sample variance is
sample variance.gif (667 bytes) or
sample var2.gif (532 bytes)

The sample standard deviation s, is
sample standard deviation formula 1.gif (847 bytes) or
sample standard deviation 2.gif (666 bytes)

Rule 3.

The sample correlation coefficient is the same as the population correlation coefficient.
sample correlation coefficient.gif (257 bytes)

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