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? AR≤ ? CR≤ ? B . Suppose that the set of all random samples of size n which are possible is X(n). De?nition 3. Any function ? A : X(n) → FT (R) is called a fuzzy statistic. Note that ? A(X1,Xn) depends only on the random sample X1,Xn and not any unknown parameters. When the observation x = (x1,xn) is given, then the value of the statistic ? A(x) is just one triangular fuzzy number. Let X be a measurable random variable on the probability space (?, F, Pr) and ? T = T(a, b, c) ∈ FT (R) be such that 2b + a + c ≥ 0. According to (2), for any ω ∈ ? we de?ne (X ? ? T)(ω) = X(ω) ? ? T.

146 APPL. COMPUT. MATH., VOL. 7, NO.1,

2007 Lemma 4. Let X be a random variable on the probability space (?, F, Pr), k1, k2 ∈ R and ? T = T(a, b, c) ∈ FT (R), where 2b + a + c ≥ 0. Then Pr(k1 ? ? TR≤X ? ? TR≤k2 ? ? T) =

1 ? α if and only if Pr(k1 ≤ X ≤ k2) =

1 ? α. Proof. See Proposition 5.1 of [11]. According to the Lemma 4, we can give the following de?nition. De?nition 4. Let ? A, ? B ∈ FT (R) be the observed fuzzy statistic, where ? AR≤ ? B. Then [ ? A, ? B] is called a 100(1 ? α)% fuzzy con?dence interval for X ? ? T, where Pr( ? AR≤X ? ? TR≤ ? B) =

1 ? α. Theorem 1. Suppose X1, X2,Xn that are independent, identically distributed random vari- ables with N(?, σ2) and U(au, bu, cu) ∈ FT (R) , L(al, bl, cl) ∈ FT (R) are the engineering fuzzy speci?cation limits, where au ≥ cl. Then the following interval is a 100(1?α)% fuzzy con?dence interval for ? Cp ? ? ∧ ? Cp ? χ2 n?1,α/2 n ?

1 , ∧ ? Cp ? χ2 n?1,1?α/2 n ?

1 ? ? (7) where ∧ ? Cp= T au?cl 6s , bu?bl 6s , cu?al 6s is the point estimation of ? Cp. Proof.See Theorem 5.1 of [11]. De?nition 5. Let [ ? A, ? B] and [ ? An, ? Bn];

n ∈ N be fuzzy intervals. De?ne i) [ ? A1, ? B1] = [ ? A2, ? B2] if [R( ? A1), R( ? B1)] = [R( ? A2), R( ? B2)], (8) ii) lim n→∞ [ ? An, ? Bn] = [ ? A, ? B] if lim n→∞ [R( ? An), R( ? Bn)] = [R( ? A), R( ? B)], (9) where lim n→∞ [R( ? An), R( ? Bn)] = lim n→∞ R( ? An), lim n→∞ R( ? Bn)]. Our observation leads us to the following theorem which was presented as an open problem in [11, 13]. Now we are ready to give the main result of this paper. Theorem 2. Assuming m[0,1] is the Lebesgue measure de?ned on [0, 1]. Under the same as- sumption as in Theorem 1, we have m[0,1] ? ? ? ? ? α;

lim n→∞ ? ? ∧ ? Cp ? χ2 n?1,α/2 n ?

1 , ∧ ? Cp ? χ2 n?1,1?α/2 n ?

1 ? ? = ∧ ? Cp , ∧ ? Cp ? ? ? ? ? = 1. (10) Proof.Since any χ2 n random variable may be written as the sum of n i.i.d. χ2 1;

so by the weak law of large numbers (Theorem 6.4.3 of [1]), we have lim n→∞ Pr χ2 n n ?

1 <

ε =

1 for every ε >

0. (11) Since g(x) = √ x is a continues function for all x >

0, by Exercise 2.2.1 of [2], it follows that lim n→∞ Pr χ2 n n ?

1 <

ε =

1 for every ε >

0. (12) Therefore, for all continuity points x of F(x), lim n→∞ Fn(x) = F(x), (13) where Fn(x) is distribution function of random variable χ2 n n and F(x) is a degenerate distrib- ution at 1. Using Lemma 1.5.6 of [17], the set {α;

lim n→∞ F?1 n (α) = F?1(α)} is at most countable A. PARCHAMI M. MASHINCHI V. PARTOVI NIA: A CONSISTENT CONFIDENCE INTERVAL

147 and hence a set of Lebesgue measure zero, where G?1(α) = infx{x;

G(x) ≥ α} is the α-quantile of the distribution function G. Hence, Theorem 1.6.3 in [17] implies that m[0,1] ? ? ? ? ? α;

lim n→∞ χ2 n,α n =

1 ? ? ? ? ? = 0, for all

0 <

α <

1. (14) Obviously, we can conclude that m[0,1] α;