Formulas as well as strategy for figures essay

Probability

Go with Law P(A) = 1 P(A)

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

P(A B) = P(A) + P(B), if A and B are contradictory

Conditional Likelihood P(A|B) sama dengan P(A B)

P(B)

Independent Condition When a and B are self-employed, P(A B) = P(A) x P(B)

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

P(A B) sama dengan P(B) by P(A|B)

Detailed Statistics

Inhabitants Mean, m= all beliefs

N

Sample Mean, times = every values

and

Population Difference, s2 = (X m)2

N

Test Variance, S2 = (x x)2

n-1

Standard Deviation = sq . root of s2 or S2

Probability Syndication

Expected Benefit, E(x) sama dengan all x P(xi = x) sama dengan m

Properties of E(x)

E(a) = a

E(ax) = aE(x)

E(ax b) = aE(x) b

E(x1 x2) sama dengan E(x1) E(x2)

E(x2) sama dengan all x2 P(xi sama dengan x)

Difference, Var(x) = E(x m)2 or Var(x) = E(x2) n(x)2

Real estate of Var(x)

Var(a) sama dengan 0

Var(ax) = a2Var(x)

Var(ax b) = a2E(x)

Var(x1 x2) = Var(x1) + Var(x2)

E(x2) = all x2 P(xi = x)

Regular Deviation = square reason behind var(x)

Binomial Distribution x ~ Rubbish bin (n, p)

Characteristics

Test consist of numerous trials

Benefits of trials are only both success or failure

Likelihood of each check between studies are the same

E(x) = np

Var(x) = npq

Ongoing Distribution back button ~ N(m, s2)

Standardising, z sama dengan x m

s

Typical Approximation to Binomial Syndication x ~ N(np, npq)

Conditions

Number of trials in >50

Must use continuity correction

Joint Likelihood

Conditional Indicate E(x | y=y1) sama dengan all x P(xi | y)

E(XY) = all x most y P(xi = back button and yi = y)

When times and y are 3rd party, E(XY) = E(X) E(Y)

Covariance of two random variables, sxy Cov(XY) = E(XY) E(X)E(Y)

Once X and Y happen to be independent, Cov(XY) = zero, since E(XY) = E(X)E(Y)

Correlation Agent, r sama dengan Cov(XY), -1 r 1

Var(x) Var(y)

Formula for Variance of linear combinations of 2 centered variables

Var(X Y) = Var(X) + Va (Y) 2Cov(XY)

Var(aX bY) = a2Var(X) + b2Var (Y) 2abCov(XY)

Distribution Of Sample Mean Sample Percentage

Let Back button denote the population variable. m the population suggest and s2 the population difference.

then

back button ~ N(m, s2/n)

Let P represent the population portion with portion P with n, the number of samples

then simply

P ~ N p , p (1-p)/n

if S is unknown

P ~ N P , P (1-P)/n approx. in which P is definitely the sample proportion with the use of continuity correction x (1/2n)

Theory Of Evaluation

Mean Sq . Error MSE = E(V q)2 where V may be the value with the estimator from your true benefit q

Greatest estimator in the true worth is the one that brings the lowest MSE

Confidence Period The time period of which the real value is definitely probable to get included.

3 Cases Of Formula Intended for Confidence Interval

Intended for population imply where

m, s2 offered, -m sama dengan x (s2/n)1/2 Zsig level

m given but s2 unknown, trials size n >50-m sama dengan x (S2/n)1/2 Zsig level

m offered but s2 unknown, samples size n < 50-m='x' (s2/n)1/2='' tsig='' level='' for='' difference='' in='' population='' means='' mx='' my='' where='' m,='' s2='' given,-='' md='(x' y)='' (sx2/nx='' +='' sy2/ny)1/2='' zsig='' level='' m='' given='' but='' s2='' unknown,='' samples='' size='' n=''>50-

mD sama dengan (x y) (Sx2/nx + Sy2/ny)1/2 Zsig level

meters given yet s2 unidentified, samples size n < 50-='' md='(x' y)='' (sp2/nx='' +='' sp2/ny)1/2='' tsig='' level='' where='' pooled='' variance,='' sp2='S(x-x)2' +='' s(y-y)2='' nx='' +='' ny='' -='' 2='' sp2='Sx2(nx-1)' +='' sy2(ny-1)='' nx='' +='' ny='' -='' 2='' paired='' samples-='' md='D' (sd2/nd)1/2='' tsig='' level='' where='' d='' is='' the='' difference='' between='' the='' paired=''>

To get Population Portion, p ~ N p, p(1-p)/n

p not really given, then it is believed with variance P(1-P)/n, in the confidence interval of

g = S (P(1-P)/n)1/2 Zsig level

Speculation Testing

Treatment:

State Null and Alternate hypothesis

Decide one or two sided test

Locate Ztest or ttest and compare the result with Zcritical and Tcritical respectively

Decision Rule, |Ztest| < zcritical='' or='' |ttest|=''>< tcritical='' then='' null='' hypothesis='' is='' true='' conclude='' in='' relation='' to='' hypothesis='' question='' e.g.,='' ztest='x' -='' m='' s/n='' p-value='' -='' decision='' rule='' reject='' h0='' if='' p-value=''>< level='' of='' significance='' accept='' h0='' if='' p-value='' level='' of='' significance=''>

Type My spouse and i Error – The mistake of rejecting H0 when H0 applies P(type My spouse and i error) = the level of relevance

Type 2 Error, w.

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