4. Suppose Yi Y, are id randonn variables with E(Y )-μ, Var(Y)= σ2 < o For...
4. Suppose Yi, Yn are iid randonn variables with E(X) = μ, Var(y)-σ2 < oo. For large n, find the approximate distribution of p = n Σηι Yi, Be sure to name any theorems you used.
4
and 5
samples, the other in small samples. Which is which? Explain. (d) Suppose we know that the 5 values are from a symmetric distribution. Then the sample median is also unbiased and consistent for the population mean. The sample mean has lower variance. Would you prefer to use the sample 4. Suppose Yi, Y, are iid r ables with E(n)-μ, Var(K)-σ2 < oo. For large n, find the approximate 5. Suppose we observe Yi...Yn from a normal distribution...
5.26 Suppose that y is N, (μ, 2), where μ LJ and -σ2ρ for all Thus E(yi-μ for all i, var(yi) 0" for all i, and cov(yoy ij; that is, the y's are equicorrelated. (a) Show that Σ can be written in the form Σ-σ2(I-P)1+a (b) Show that Σ-i(vi-y?/(r2(1-p] is X2(n-1)
5.26 Suppose that y is N, (μ, 2), where μ LJ and -σ2ρ for all Thus E(yi-μ for all i, var(yi) 0" for all i, and cov(yoy ij; that...
4. The moment generating function of the normal distribution with parameters μ and σ2 is (t) exp ( μ1+ σ2t2 ) for -oo < t oo. Show that E X)-ψ(0)-μ and Var(X)-ψ"(0)-[ty(0)12-σ2. 5. Suppose that X1, X2, and X3 are independent random variables such that E[X]0 and ElX 1 for i-12,3. Find the value of E[LX? (2X1 X3)2] 6. Suppose that X and Y are random variables such that Var(X)-Var(Y)-2 and Cov(X, Y)- 1. Find the value of Var(3X -...
S y, and that yi-μ +Ei. You can assume that Ele]-0 for all i, Ele: -σ2 for all i, and Ele#3-0 for all i j You want to estimate a sample mean, and your friend tells you to use the following estimator: uppose that vou have collected n observations on where w is a known sample weight for observation i (this means w; is non-random) (a) Find E( (b) Under what conditions, if any, is p an unbiased estimator? Under...
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?
4. Let X1,X2, ,Xn be a randonn sample from N(μ, σ2) distribution, and let s* Ση! (Xi-X)2 and S2-n-T Ση#1 (Xi-X)2 be the estimators of σ2 (i) Show that the MSE of s is smaller than the MSE of S2 (ii) Find E [VS2] and suggest an unbiased estimator of σ.
1. Let X1, X2, , Xn be independent Normal μ, σ2) random variables. Let y,-n Σ_lx, denote a sequence of random variables (a) Find E(y,) and Var(y,) for all n in terms of μ and σ2. (b) Find the PDF for Yn for alln. (c) Find the MGF for Yn for all n.
2. Suppose the variables Yi and Y have the following properties EQİ)-4, Var(h)-19, E(Y )-6.5, Var(Ya)-5.25, E(Y3%)-30 Calculate the following; please show the underlying work a) (3 pts) Cov(, ) b) (3 pts) Cov(41, 3%) c) (3 pts) Cov(41.5-½) (6 pts) Find the correlation coefficient between 1 + 3, and 3-2%
Let X and Y be random variables with the follow E(Y) μ,--2 Var(x) o, 0.3 Var(Y)-σ,-0.5 Cov(XY) o,,-0.03 Find the following: ESX-3 Y)