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# Extra - Chapter 1 Descriptive Statistics Sample Average 1...

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Unformatted text preview: Chapter 1: Descriptive Statistics Sample Average _ 1 2:;1mi 117:; Population Average N ,U : ﬁ 21:1 5“ 8 2—R:IZ(\$i E)2_n:1(2wf (Sexy) 11 1 . . Pk = _ E devrations (pk) of Chebychev’s Rule: The proportion of observations that are within 3:: standard the mean is at least: hapter 2: Probability Multiplication rule Permutation (k. — —’— Combination For anv two events A and B: __ pm UB):P(A)+P(B)— P{An B) n (rt—E)! R1 Xﬂgx'ﬂz...xnk Plan: Two events A and B are independent if conditional probability of A given P(A|B) = P(A) Chapter 3:Discrete PDF E(aX + b) : eE(X] +2: wax + b) = (Bl/(X) = £1,202 3.1 Binomial Distribution For X N binomial(n, p) a : ﬁxed number of trials p : probability of succes (S) r = number of successes (S) M =m)=(:)p=(1—p)H r:0,1,2,...,n # =ElX] =11? U2 =ViXi =E[(£—#)2]= 19(1 —pl that B occurred [P(B) > 0): HA and B are independent then P{AlB) _ P(A r1 B) PM n B) : PtAiPtB) - —p(3) 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, . . up...) n = Number of trials. r = Number of possible outcomes. p.- : P (Outcome 2‘ on any particular trial}. r;- = Number of trials resulting in outcome 3‘. n! I ,- _ 9:11:32!” .rripflpfmp: r1+rg+...r,:r p(mlim23 ' ' ' 1231') mi=0,1,2,... .6 Negative Binomial Probability Distribution | or X N negative binomial[r,p} 9. P ELK] = p, : number of S V X i 2 = probability of S i l T J _ rtl-P) , rtl-P) —2— P p 112' = the number of failures preceding the r‘th success If r = 1 we have a Geometric distribution. 3.7 Poisson Distribution E[X] = p. = A For X N poissonOt) V[X] = 0'2 = A 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 e‘AAz 5r! A>0 P(X=:'BJ= £320,132,... 3.8 Poisson Approximation to the Binomial Distribution Let X be a binomial random variable with probability distribution X m binomial(n, p). Whenn—rooandp—vﬂandA2np remains ﬁxed at A > D, then X N binomial[n,p) —> X N poisson()\ 2 up” hapter 4: Continuous PDF’S Pm. g X 5 b) = F(b) — Fm) ngx gap/biceps FTX) —f(\$) M) = P(X : m) = f randy For 0 g p g 1 the (lﬂﬂpl’th percentile of a continuous Em =.u = If; (X— pJZ] : 0'2 : Links — u)2 -f(:r)d:r Erxem =EiX217Ei r = 02 E distribution you must solve p = F (2:) for a: where a is the (100p)’th percentile. 4.1 The Uniform Distribution The family of uniform distributions has the following PDF: f 4.1.1 The Exponential Distribution The family of exponential distributions has the following PDF: :1: 2 O, A > 0 otherwise aw) : { ...
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