Suppose x ~ Multinomial (n,p1...,pn). Determine μ = E[x] AND Σ = var[x]
Suppose x ~ Multinomial (n,p1...,pn). Determine μ = E[x] AND Σ = var[x]
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...
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...
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...
1. Suppose (x, Y) has bivariate normal distribution, E(x) E(Y)- 0, Var(X) σ , Var(Y) σ and Correl(X, Y) p. Calculate the conditional expectation E(X2|Y).
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)
2 0 -1 EIX) = μ = | 0 | and var(X) Σ _ | 0 -1 0.5 3 0.5 | compute: (a) E[Xi +Xs] (c) var(X2- X3) d var(X2 + X) (e) cov(4X2 +X1,3Xi -2X)
2 0 -1 EIX) = μ = | 0 | and var(X) Σ _ | 0 -1 0.5 3 0.5 | compute: (a) E[Xi +Xs] (c) var(X2- X3) d var(X2 + X) (e) cov(4X2 +X1,3Xi -2X)
Problem 4 Suppose X ~N(0, 1) (1) Explain the density of X in terms of diffusion process. (2) Calculate E(X), E(X2), and Var(X). (3) Let Y = μ +ơX. Calculate E(Y) and Var(Y). Find the density of Y.
Problem 4 Suppose X ~N(0, 1) (1) Explain the density of X in terms of diffusion process. (2) Calculate E(X), E(X2), and Var(X). (3) Let Y = μ +ơX. Calculate E(Y) and Var(Y). Find the density of Y.
Consider a random variable X ~ N(μ, σ), where both μ and σ are unknown. Suppose we have n 1.1.0. samples generated from X. How do we construct a 95% confidence interval? Consider the cases n-: 10 and 1000. Use simulation to validate this confidence interval.
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.