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2. Suppose that ξι, ξ2, . . . are 1.1.d. RVs with Εξι-μ and Var (6)-σ2 E (0,00). Set X-3kE+2,1,2,, and let Sn X+Xn,...
3. Let X be a continuous random variable with E(X)-μ and Var(X)-σ2 < oo. Suppose we...
3. Let X be a continuous random variable with E(X)-μ and Var(X)-σ2 < oo. Suppose we try to estimate μ using these two estimators from a random sample X, , X,: For what a and b are both estimators unbiased and the relative efficiency of μι to is 45n?
Let p0 =P(X=1) and suppose that 0<p0 <1. Let μ=E(X) and σ2 =var(X). a.) Find E[X|X...
Let p0 =P(X=1) and suppose that 0<p0 <1. Let μ=E(X) and σ2 =var(X). a.) Find E[X|X ̸= 1] b.) Find var(X|X ̸= 1)
R1. Suppose X is a continuous RV with E(X-μ and Var(X-σ2 where both μ and σ...
R1. Suppose X is a continuous RV with E(X-μ and Var(X-σ2 where both μ and σ are unknown. Note that X may not be a normal distribution. Show that X is an asymptotically unbiased estimator for μ2. (This problem does not require the computer.) R2. Let X ~ N(μ 10.82). Following up on R1, we will be approximating μ2, which we can see should be 100, For now, let the sample size be n 3. Pick 3 random numbers from...
4) Let Xi , X2, . . . , xn i id N(μ, σ 2) RVs. Consider the problem of testing Ho : μ- 0 against H1: μ > 0. (a) It suffices to restrict attention to sufficient statistic (U, v), where U X and V...
4) Let Xi , X2, . . . , xn i id N(μ, σ 2) RVs. Consider the problem of testing Ho : μ- 0 against H1: μ > 0. (a) It suffices to restrict attention to sufficient statistic (U, v), where U X and V S2. Show that the problem of testing Ho is invariant under g {{a, 1), a e R} and a maximal invariant is T = U/-/ V. (b) Show.that the distribution of T has MLR,...
RI. Suppose X is a continuous RV with E(X)-μ and Var(X)-σ2 where both μ and σ...
RI. Suppose X is a continuous RV with E(X)-μ and Var(X)-σ2 where both μ and σ are unknown. Note that X may not be a normal distribution. Show that X is an asymptotically unbiased estimator for μ. (This problem does not require the computer.) R2. Let X ~ ŅĢi-10.82). Following up on RI, we will be approximating μ2, which we can see should be 100. For now, let the sample size be n = 3, Pick 3 random numbers from...
2. Let Xi,... Xn be a random sample from the density f(x:0) 1o otherwise Suppose n...
2. Let Xi,... Xn be a random sample from the density f(x:0) 1o otherwise Suppose n = 2m+1 for some integer m. Let Y be the sample median and Z = (a) Apply the usual formula for the density of an order statistic to show the density max(X1) be the sample maximum. of Y is 0) 6 3) (b) Note that a beta random variable X has density re+ β22 a-1 (1-2)8-1 with mean μ α/G + β) and variance...
Please ignore part abc 4. Suppose that (X1, Yİ), , (XN,Yv) denotes a random sample. Let Si = a + bX, T, e+ dy, where a, b, c and d are constants. Let X ΣΧ, and σ2-NL Σ(x,-x)2, with the analogous expr...
Please ignore part abc
4. Suppose that (X1, Yİ), , (XN,Yv) denotes a random sample. Let Si = a + bX, T, e+ dy, where a, b, c and d are constants. Let X ΣΧ, and σ2-NL Σ(x,-x)2, with the analogous expressions for y S, T. Let σΧΥ-ΝΤΣ (Xi-X)(X-Y), and let P:XY ƠXY/(ƠXƠY), with the analogous expressions for S, T. (a) Show that σ bbe (b) Show that ớsı, d ớx (c) Show that psT ST (d) How do the...
3. Let X be a continuous random variable with E(X)-μ and Var(X)-σ2 < oo. Suppose we try to estimate μ using these two estimators from a random sample X, , X,: For what a and b are both estimators unbiased and the relative efficiency of μι to is 45n?
R1. Suppose X is a continuous RV with E(X-μ and Var(X-σ2 where both μ and σ are unknown. Note that X may not be a normal distribution. Show that X is an asymptotically unbiased estimator for μ2. (This problem does not require the computer.) R2. Let X ~ N(μ 10.82). Following up on R1, we will be approximating μ2, which we can see should be 100, For now, let the sample size be n 3. Pick 3 random numbers from...
4) Let Xi , X2, . . . , xn i id N(μ, σ 2) RVs. Consider the problem of testing Ho : μ- 0 against H1: μ > 0. (a) It suffices to restrict attention to sufficient statistic (U, v), where U X and V S2. Show that the problem of testing Ho is invariant under g {{a, 1), a e R} and a maximal invariant is T = U/-/ V. (b) Show.that the distribution of T has MLR,...
RI. Suppose X is a continuous RV with E(X)-μ and Var(X)-σ2 where both μ and σ are unknown. Note that X may not be a normal distribution. Show that X is an asymptotically unbiased estimator for μ. (This problem does not require the computer.) R2. Let X ~ ŅĢi-10.82). Following up on RI, we will be approximating μ2, which we can see should be 100. For now, let the sample size be n = 3, Pick 3 random numbers from...
2. Let Xi,... Xn be a random sample from the density f(x:0) 1o otherwise Suppose n = 2m+1 for some integer m. Let Y be the sample median and Z = (a) Apply the usual formula for the density of an order statistic to show the density max(X1) be the sample maximum. of Y is 0) 6 3) (b) Note that a beta random variable X has density re+ β22 a-1 (1-2)8-1 with mean μ α/G + β) and variance...
Please ignore part abc
4. Suppose that (X1, Yİ), , (XN,Yv) denotes a random sample. Let Si = a + bX, T, e+ dy, where a, b, c and d are constants. Let X ΣΧ, and σ2-NL Σ(x,-x)2, with the analogous expressions for y S, T. Let σΧΥ-ΝΤΣ (Xi-X)(X-Y), and let P:XY ƠXY/(ƠXƠY), with the analogous expressions for S, T. (a) Show that σ bbe (b) Show that ớsı, d ớx (c) Show that psT ST (d) How do the...
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