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# Extra Notes 4.png - Chapter 1 Descriptive Statistics Sample...

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Unformatted text preview: Chapter 1: Descriptive Statistics Sample Average Population Average _ N f2§23=1mi #:%Z,—:1ri 82 _n:12(\$i E)2_n:1(2wf (21:31)) Pk: —E Chebychev’s Rule: The proportion of observations that are within 3:: standard 1 deviations (pk) of the mean is at least: hapter 2: Probability Multiplication rule Permutation Combination For anv two events A and B: _ P(A U B) :P(A)+P(B) — P{A n B) n1 xngxnz...xnk Pkﬁzﬁ Two events A and B are independent if conditional probability of A given that B occurred [P(B) > 0): P(A|B) = P(A) HA and B are independent then Chapter 3:Discrete PDF E(aX + b) : cE(X) +b V(a.X + b) = cﬂ/(X) = azai’ 3.1 Binomial Distribution For X N binomial“;J p} n : ﬁxed number of trials p : probability of succes (S) :r : number of successes (S) Tl. P(X =3)=(m)pz(1,p)n—= :r:0,1,2,...,n PA B PtAnBJ:P(A)P(B) PWBJ=% n=ElX1 =71? 2=Vle=EH\$—#)21=HP(1—P . If A, B, C, D, . . . are mutually independent then P(AﬂB N Go D. . .) : P(A)P(B)P(C)P(D) . . 3.2 Multinomial Distribution For X N multinomial(n,p1, . . . ,pr) 11 = Number of trials. 5“ = Number of possible outcomes. .6 Negative Binomial Probability Distribution | or X N negative binomial[r,p} 'r : number of S p = probability of S Em = a vmp:ﬁ _ rﬂ-P) P , rﬂ-P) 7 P2 a: = the number of failures preceding the r‘th success If 'r = 1 we have a Geometric distribution. 3.7 Poisson Distribution E[X] = ,u = A For X N poissonOi) vmp=ﬂ=i A = the rate per unit time or rate per unit area. m = the number of successes occurring during a given time interval or in a speciﬁed region When n _’ 00 and P _' 0 and A 2 up Pg=m= e‘AAz 9:! \$20,1,2,... A>0 3.8 Poisson Approximation to the Binomial Distribution Let X be a binomial random variable with probability distribution X m binomialm, p). remains ﬁxed at A > D, then X N binomial[n,p) —> X N poisson()\ 2 up” pg- : P(Outcome 2‘ on any particular trial). m:- = Number of trials resulting in outcome 2‘. n! p(ml‘l \$27 ' ' ' 7 \$1") : Ellﬂigl... m1+2¢2+...:r,=r azanam m, masks r 1" ...
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