Term Papers on Formulas

Term Paper Title	Formulas
# of Words	1280
# of Pages (250 words per page double spaced)	5.12

Formulas

FORMULAS / STRATEGY FOR STATISTICS

Probability

Complement Law - P(A’) = 1 - P(A)

Laws Of Addition -     P(A È B) = P(A) + P(B) - P(A Ç B), if A and B not mutually exclusive
     P(A È B) = P(A) + P(B), if A and B are mutually exclusive

Conditional Probability - P(A|B) = P(A Ç B)
     P(B)

Independent Condition - If A and B are independent, P(A Ç B) = P(A) x P(B)

Laws Of Multiplication - If A and B are dependent, P(A Ç B) = P(A) x P(B|A) or
     P(A Ç B) = P(B) x P(A|B)

Descriptive Statistics

     Population Mean, m= å all values
     N

Sample Mean, x’ = å all values
     n

Population Variance, s2 = å (X - m)2
     N

Sample Variance, S2 = å (x - x’)2
     n-1

Standard Deviation = square root of s2 or S2

Probability Distribution

Expected Value, E(x) = å all x P(xi = x) = m
Properties of E(x),
E(a) = a
E(ax) = aE(x)
E(ax ± b) = aE(x) ± b
E(x1 ± x2) = E(x1) ± E(x2)
E(x2) = å all x2 P(xi = x)

Variance, Var(x) = E(x - m)2 or Var(x) = E(x2) - n(x’)2
Properties of Var(x),
Var(a) = 0
Var(ax) = a2Var(x)
Var(ax ± b) = a2E(x)
Var(x1 ± x2) = Var(x1) + Var(x2)
E(x2) = å all x2 P(xi = x)

Standard Deviation = square root of var(x)

Binomial Distribution - x ~ Bin (n , p)
Characteristics,
Experiment consist of a number of trials
Results of trials are only either success or failure
Probability of each test between trials are the same

E(x) = np
     Var(x) = npq

Continuous Distribution - x ~ N(m , s2)
     Standardising, z = x - m
     s

Normal Approximation to Binomial Distribution - x ~ N(np , npq)
Conditions,
Number of trials n > 50
Must use continuity correction

Joint Probability

Conditional Mean - E(x | y=y1) = å all x P(xi | y)

E(XY) = å [all x all y P(xi = x and yi = y)]
When x and y are independent, E(XY) = E(X) E(Y)

Covariance of 2 random variables, sxy - Cov(XY) = E(XY) - E(X)E(Y)
     When X and Y are independent, Cov(XY) = 0, since E(XY) = E(X)E(Y)

Correlation Coefficient, r =      Cov(XY)          ,-1 £ r £ 1
     Ö[Var(x) Var(y)]

Formula for Variance of linear combinations of 2 dependent variables -
Var(X ± Y) = Var(X) + Var (Y) ± 2Cov(XY)
Var(aX ± bY) = a2Var(X) + b2Var (Y) ± 2abCov(XY)

Distribution Of Sample Mean Sample Proportion

Let X denote the population variable. m the population mean and s2 the population variance.
then,
x’ ~ N(m,s2/n)

Let P denote the population proportion with proportion P with n, the number of samples,
then
P ~ N { p , p [(1-p)/n] }

if P is unknown,
P ~ N { P , P [(1-P)/n] } approx. where P is the sample proportion with the use of continuity correction x ± (1/2n)

Theory Of Estimation

Mean Square Error - MSE  = E(V - q)2 where V is the value of the estimator from the true value q
     Best estimator of the true value is the one that yields the lowest MSE

Confidence Interval - The interval of which the true value is probable to be included.

3 Cases Of Formula For Confidence Interval -

For population mean where
m, s2 given,     -     m = x’ ± (s2/n)1/2 Zsig level
m given but s2 unknown, samples size n > 50     -     m = x’ ± (S2/n)1/2 Zsig level
m given but s2 unknown, samples size n  50     -
mD = (x’ ± y’) ± (Sx2/nx + Sy2/ny)1/2 Zsig level

m given but s2 unknown, samples size n < 50     -
mD = (x’ ± y’) ± (Sp2/nx + Sp2/ny)1/2 tsig level where pooled variance, Sp2 = S(x...

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