Midterm 1

# Midterm 1 - I.05 method(i-0.5)n*100=k for kth%ile Returns...

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I+.05 method: ( - . ) * = i 0 5 n 100 k for kth %ile. Returns item place in array of ordered set. K as integer. N+1 method: + = kn 1 i . K in decimal form. For interpolators (e.g. 3.5), ipart + fpart(larger-smaller). Sample mean is just average Sample variance = = ( - ) - s2 i 1n xi x 2n 1 PDFs: P(a<x<b) = abfxdx 1) F(x) >= 0 i.e. = fx kx2 < < , for 0 x 2 0 elsewhere 2) -∞ ∞ ( ) f x = 1 3) CDF: 4) F(x) = P(X<x) = -∞ xfudu . Shows amount of probability to that point. 5) 6) f(x) = some PDF 7) µ=E[x]=-∞ xfxdx 8) = =-∞∞( - ) = -∞∞ - σ2 Vx x μ 2fxdx x2fxdx μ2 9) Normal Distribution: 10) X:n(µ, σ 2 ) -> = - Z x μσ 11) P(Z>z)=P(Z<-z) 12) 13) PMFs: 1) F(x) >= 0 for all x 2) ( ) f x =1 3) = = ( ) μ Ex xf x 4) = = - σ2 VX Ex2 μ2 5) = ≤ =-∞ = < , = ≤ ≤ , . F PX x xfx Fx 0 for x 1 Fx 18 for 1 x 3 etc 6) Binomial distribution (discrete): 7) Deals with some # of trials with one of two outcomes (success or failure in general) 8) : , → = ( ) ( - ) - , = , , ,… X bn p fx xn px 1 p n x x 0 1 2 n 9) E[x]=n-p 10) V(x)=np(1-p) 11) CDF of: 0xpdf 12) Poisson Distribution (discrete): 13) P(X=x)= - ! e λλxx 14) X:p(x, λ), and x is # of some event in some time/distance 15) Poisson is for some defined interval with events occurring at random 16) Assure events are random through the interval, and can be split into subintervals so that: 1) Probability of more than one count in a subinterval is 0 2) Prob. Of one count of subinterval is same for all, and proportional to length 3) Count in each sub. Is independent of other subintervals. 17) Exponential Distribution (continuous): 18) Timed, so not binomial/poisson 19) Time between successive events of a Poisson process 20) PDF: - λe λx for ≤ ≤∞ 0 x 21) Good for slow wear-out 22) Has lack of memory 23) E(X) = , V(x)= 1λ2 24) e.g. X=distance until first crack, and need to find probability of no cracks in 10-mile stretch 25) P(X>10)= - =- - 10 λe λx e λx|10 26) If there are a # of cracks in 10 miles, use Poisson 27)

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28) Joint PDFs: 29) F(x,y)->P(a<x<b, c<Y<d) = ( , ) abcdf x y dxdy 30) 31) Let Y = x+c 32) E[Y]=E(X+c), E[x]=µ 33) =E[x]+E[c]->µ+c 34) V(Y)=V(X)+V(c)=V(X)=σ 2 35) Y=cx 1 , E[Y]=cE[x]=cµ 36) V(Y)=c 2 V(X) = c 2 σ 2 37) 38) Y=c 1 x 1 +c 2 x 2 +…c n x n 39) E(Y)=c 1 µ 1 +c 2 µ 2 +…+c n µ n 40) V(Y)=c 1 2 σ 1 2 +c 2 2 σ 2 2 41) 42) Normal Prob. Plots 1) Rank data in ascending order 2) Plot each pt with i-.5 method (P=(i-0.5)/n 43) 44) E(A-B) = E(A)-E(B) 45) V(A+-B) = V(A)+V(B) 46) e.g. 47) A:n(stuff) 48) B:n(stuff) 49) C:n(stuff) 50) D=A-B-C 51) E(D) = µ A B C 52) V(D) = σ A 2 B 2 C 2 53) 54) Central Limit Theorem: 55) If underlying population is unknown, 56) X1,x2…xn R.S. (mu, sigma squared), and if x-bar is sampling mean, 57) Then = - Z x μσn is n(0, 1) as n-> infinity 58) Type I error: Rejecting H 0 when it is true. Type II error: Failing to reject H 0 when it is false. 59) α=P(type I error) = P(reject H
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## This note was uploaded on 09/17/2008 for the course IEE 380 taught by Professor Anderson-rowland during the Spring '06 term at ASU.

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Midterm 1 - I.05 method(i-0.5)n*100=k for kth%ile Returns...

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