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Let X1,.-. , Xn ~ N(2, 1) be independent, where E R is unknown. (i) Show...
2. Let X.... , X, N(0,%) be independent, where a > 0 is unknown. (1) Is...
2. Let X.... , X, N(0,%) be independent, where a > 0 is unknown. (1) Is 52:= -- .(X- X) sufficient? If yes, is it minimum sufficient? (ii) Is T :=1 X sufficient? If yes, is it minimum sufficient? (iii) Show that and are ancillary. (iv) Is 52 or T complete? If yes, verify directly that it is independent of and (Hint: for the independence, use the joint distribution of (X,s) and then use transformations.)
May 21, 2019 R 3+3+5-11 points) (a) Let X1,X2, . . Xn be a random sample from G distribution. Show that T(Xi, . . . , x,)-IT-i xi is a sufficient statistic for a (Justify your work). (b) Is Unifo...
May 21, 2019 R 3+3+5-11 points) (a) Let X1,X2, . . Xn be a random sample from G distribution. Show that T(Xi, . . . , x,)-IT-i xi is a sufficient statistic for a (Justify your work). (b) Is Uniform(0,0) a complete family? Explain why or why not (Justify your work) (c) Let X1, X2, . .., Xn denote a random sample of size n >1 from Exponential(A). Prove that (n - 1)/1X, is the MVUE of A. (Show steps.)....
Let X1, X2, , xn are independent random variables where E(X)-? and Var(X) ?2 for all...
Let X1, X2, , xn are independent random variables where E(X)-? and Var(X) ?2 for all i = 1, 2, , n. Let X-24-xitx2+--+Xy variables. is the average of those random Find E(X) and Var(X).
2. Suppose that X1, X2, .. , Xn are iid N(0, 02). Where i and o both assumed to be unknown. Let 0 = (i,a). Find jointly...
2. Suppose that X1, X2, .. , Xn are iid N(0, 02). Where i and o both assumed to be unknown. Let 0 = (i,a). Find jointly sufficient statistics for 0
4. Let X1, X2, . .. be independent random variables satisfying E(X) E(Xn) --fi. (a) Show...
4. Let X1, X2, . .. be independent random variables satisfying E(X) E(Xn) --fi. (a) Show that Y, = Xn - E(Xn) are independent and E(Yn) = 0, E(Y2) (b) Show that for Y, = (Y1 + . . + Y,)/n, <B for some finite B > 0 and VB,E(Y) < 16B. 16B 6B 1 E(Y) E(Y) n4 i1 n4 n3 (c) Show that P(Y, > e) < 0 and conclude Y, ->0 almost surely (d) Show that (i1 +...
In 10. 11, Let X1, X2, . , Xn and Yi, Y2, . . . , Y,, be independent samples from N(μ, σ?) and N(...
In 10. 11, Let X1, X2, . , Xn and Yi, Y2, . . . , Y,, be independent samples from N(μ, σ?) and N(μ, σ), respectively, where μ, σ. ơỈ are unknown. Let ρ-r/of and g m/n, and consider the problem of unbiased estimation of u
In 10. 11, Let X1, X2, . , Xn and Yi, Y2, . . . , Y,, be independent samples from N(μ, σ?) and N(μ, σ), respectively, where μ, σ. ơỈ are unknown....
2-3. Let ?>0 and ?? R. Let X1,X2, distribution with probability density function , Xn be...
2-3. Let ?>0 and ?? R. Let X1,X2, distribution with probability density function , Xn be a random sample from the zero otherwise suppose ? is known. ( Homework #8 ): W-X-5 has an Exponential ( 2. Recall --)-Gamma ( -1,0--) distribution. a) Find a sufficient statistic Y-u(X1, X2, , Xn) for ? b) Suggest a confidence interval for ? with (1-?) 100% confidence level. "Flint": Use ?(X,-8) ? w, c) Suppose n-4, ?-2, and X1-215, X2-2.55, X3-210, X4-2.20. i-1...
1. Let X1, ... , Xn be independent random variables taking values 0 or 1 with...
1. Let X1, ·s, Xn be independent random variables taking values 0 or 1 withP(Xi=1)=eθ-ai /(1+eθ-ai ), i=1, ……, nfor some given constants ai. Find a one-dimensional sufficient statistic for θ.
5. Let Xi,..., X, be iid N(e, 1). (a) Show that X is a complete sufficient statistic. (b) Show th...
1.(c)
2.(a),(b)
5. Let Xi,..., X, be iid N(e, 1). (a) Show that X is a complete sufficient statistic. (b) Show that the UMVUE of θ 2 is X2-1/n x"-'e-x/θ , x > 0.0 > 0 6. Let Xi, ,Xn be i.i.d. gamma(α,6) where α > l is known. ( f(x) Γ(α)θα (a) Show that Σ X, is complete and sufficient for θ (b) Find ElI/X] (c) Find the UMVUE of 1/0 -e λ , X > 0 2) (x...
Minimum and maximum of n independent exponentials. Let X1, X2, ..., Xn be independent, each with...
Minimum and maximum of n independent exponentials. Let X1, X2, ..., Xn be independent, each with exponential (~) distribution. Let V min (X1, X2, ..., Xn) and W = max(X1, X2, ..., Xn). Find the joint density of V and W. .
2. Let X.... , X, N(0,%) be independent, where a > 0 is unknown. (1) Is 52:= -- .(X- X) sufficient? If yes, is it minimum sufficient? (ii) Is T :=1 X sufficient? If yes, is it minimum sufficient? (iii) Show that and are ancillary. (iv) Is 52 or T complete? If yes, verify directly that it is independent of and (Hint: for the independence, use the joint distribution of (X,s) and then use transformations.)
May 21, 2019 R 3+3+5-11 points) (a) Let X1,X2, . . Xn be a random sample from G distribution. Show that T(Xi, . . . , x,)-IT-i xi is a sufficient statistic for a (Justify your work). (b) Is Uniform(0,0) a complete family? Explain why or why not (Justify your work) (c) Let X1, X2, . .., Xn denote a random sample of size n >1 from Exponential(A). Prove that (n - 1)/1X, is the MVUE of A. (Show steps.)....
Let X1, X2, , xn are independent random variables where E(X)-? and Var(X) ?2 for all i = 1, 2, , n. Let X-24-xitx2+--+Xy variables. is the average of those random Find E(X) and Var(X).
2. Suppose that X1, X2, .. , Xn are iid N(0, 02). Where i and o both assumed to be unknown. Let 0 = (i,a). Find jointly sufficient statistics for 0
4. Let X1, X2, . .. be independent random variables satisfying E(X) E(Xn) --fi. (a) Show that Y, = Xn - E(Xn) are independent and E(Yn) = 0, E(Y2) (b) Show that for Y, = (Y1 + . . + Y,)/n, <B for some finite B > 0 and VB,E(Y) < 16B. 16B 6B 1 E(Y) E(Y) n4 i1 n4 n3 (c) Show that P(Y, > e) < 0 and conclude Y, ->0 almost surely (d) Show that (i1 +...
In 10. 11, Let X1, X2, . , Xn and Yi, Y2, . . . , Y,, be independent samples from N(μ, σ?) and N(μ, σ), respectively, where μ, σ. ơỈ are unknown. Let ρ-r/of and g m/n, and consider the problem of unbiased estimation of u
In 10. 11, Let X1, X2, . , Xn and Yi, Y2, . . . , Y,, be independent samples from N(μ, σ?) and N(μ, σ), respectively, where μ, σ. ơỈ are unknown....
2-3. Let ?>0 and ?? R. Let X1,X2, distribution with probability density function , Xn be a random sample from the zero otherwise suppose ? is known. ( Homework #8 ): W-X-5 has an Exponential ( 2. Recall --)-Gamma ( -1,0--) distribution. a) Find a sufficient statistic Y-u(X1, X2, , Xn) for ? b) Suggest a confidence interval for ? with (1-?) 100% confidence level. "Flint": Use ?(X,-8) ? w, c) Suppose n-4, ?-2, and X1-215, X2-2.55, X3-210, X4-2.20. i-1...
1. Let X1, ·s, Xn be independent random variables taking values 0 or 1 withP(Xi=1)=eθ-ai /(1+eθ-ai ), i=1, ……, nfor some given constants ai. Find a one-dimensional sufficient statistic for θ.
1.(c)
2.(a),(b)
5. Let Xi,..., X, be iid N(e, 1). (a) Show that X is a complete sufficient statistic. (b) Show that the UMVUE of θ 2 is X2-1/n x"-'e-x/θ , x > 0.0 > 0 6. Let Xi, ,Xn be i.i.d. gamma(α,6) where α > l is known. ( f(x) Γ(α)θα (a) Show that Σ X, is complete and sufficient for θ (b) Find ElI/X] (c) Find the UMVUE of 1/0 -e λ , X > 0 2) (x...
Minimum and maximum of n independent exponentials. Let X1, X2, ..., Xn be independent, each with exponential (~) distribution. Let V min (X1, X2, ..., Xn) and W = max(X1, X2, ..., Xn). Find the joint density of V and W. .
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