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Sampling
Distribution Theory
ch6
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F Distribution: F(r1, r2)
 From two indep. random samples of size n1 & n2 from
N(μ1,σ12) & N(μ2,σ22),some comparisons can be performed,
such as:
 Fα(r1,r2)is the upper 100α percent point. [Table VII, p.689~693]
 Ex: Suppose F has a F(4,9) distribution.
 Find constants c & d, such that P(F c)=0.01, P(F d)=0.05
c=F0.99(4,9)
If F is F(6,9):
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Order Statistics
 The order statistics are the observations of the random sample arranged
in magnitude from the smallest to the largest.
 Assume there is no tie: identical observations.
 Ex6.9-1: n=5 trials: {0.62, 0.98, 0.31, 0.81, 0.53} for the p.d.f.
f(x)=2x, 0<x<1. The order statistics are {0.31, 0.53, 0.62, 0.81, 0.98}.
 The sample median is 0.62, and the sample range is 0.98-0.31=0.67.
 Ex6.9-2: Let Y1<Y2<Y3<Y4<Y5 be the order statistics for X1, X2, X3,
X4, X5, each from the p.d.f. f(x)=2x, 0<x<1.
 Consider P(Y4<1/2) ≡at least 4 of Xi’s must be less than 1/2: 4
successes.
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General Cases
 The event that the rth order statistic Yr is at most y, {Yr≤y}, can occur iff at
least r of the n observations are no more than y.
 The probability of “success” on each trial is F(y).
 We must have at least r successes. Thus,
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Alternative Approach
 A heuristic approach to obtain gr(y):
 Within a short interval Δy:
 There are (r-1) items fall less than y, and (n-r) items above y+Δy.
On a single trial
 The multinomial probability with n trials is approximated as.
 Ex6.9-3: (from Ex6.9-2) Y1<Y2<Y3<Y4<Y5 are the order statistics for X1,
X2, X3, X4, X5, each from the p.d.f. f(x)=2x, 0<x<1.
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More Examples
 Ex: 4 indep. Trials(Y1 ~ Y4) from a distribution with f(x)=1,
0<x<1.
 Find the p.d.f. of Y3.
 Ex: 7 indep. trials(Y1 ~ Y7) from a distribution f(x)=3(1-x)2,
0<x<1.
 Find the p.d.f. of the sample median, i.e. Y4, is less than
 Method 1: find g4(y), then
 Method 2: find
then
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By Table II on p.677.
Order Statistics of Uniform Distributions
 Thm3.5-2: if X has a distribution function F(X), which has
U(0,1).
{F(X1),F(X2),…,F(Xn)}
⇒Wi’s are the order statistics of n indep. observations from U(0,1).
 The distribution function of U(0,1) is G(w)=w, 0<w<1.
 The p.d.f. of the rth order statistic Wr=F(Yr) is
p.d.f. Beta
⇒Y’s partition the support of X into n+1 parts, and thus
n+1 areas under f(x) and above the x-axis.
 Each area equals 1/(n+1) on the average.
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Percentiles
 The (100p)th sample percentile πp is defined s.t. the area under
f(x) to the left of πp is p.
 Therefore, Yr is the estimator of πp, where r=(n+1)p.
 In case (n+1)p is not an integer, a (weighted) average of Yr and Yr+1 can
be used, where r=floor[(n+1)p].
 The sample median is
 Ex6.9-5: X is the weight of soap; n=12 observations of X is listed:
 1013, 1019, 1021, 1024, 1026, 1028, 1033, 1035, 1039, 1040, 1043, 1047.
 ∵n=12, the sample median is
 ∵(n+1)(0.25)=3.25, the 25th percentile or first quartile is
 ∵(n+1)(0.75)=9.75, the 75th percentile or third quartile is
 ∵(n+1)(0.6)=7.8, the 60th percentile
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Another Example
 Ex5.6-7: The order statistics of 13 indep. Trials(Y1<Y2< …<
Y13) from a continuous type distribution with the 35th
percentile π0.35.
 Find P(Y3< π0.35< Y7)
Success
 The event {Y3< π0.35< Y7} happens iff there are at least 3 but less
than 7 “successes”, where the success probability is p=0.35.
By Table II on p.677~681.
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