Let X=(X1,…,Xn)′ be the n×p data matrix, where Xi=(Xi1,…,Xip)′ is the ith observation. Let X¯=n−1∑ni=1Xi be...
Let X=(X1,…,Xn)′ be the n×p data matrix, where Xi=(Xi1,…,Xip)′ is the ith observation. Let X¯=n−1∑ni=1Xi be the sample mean. Let sj1j2=1/n∑ni=1(Xij1−X¯j1)(Xij2−X¯j2) be the sample covariance between the j1th and j2th variables. Let S=(sj1j2) be the sample covariance matrix. Show that S=1nX′X−X¯′X¯.
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Let X=(X1,…,Xn)′ be the n×p data matrix, where Xi=(Xi1,…,Xip)′
is the ith observation. Let X¯=n−1∑ni=1Xi be...
Let Xi, ..., Xn be random variables with the same mean and with covariance function where...
Let Xi, ..., Xn be random variables with the same mean and with covariance function where |ρ| < 1 . Find the mean and variance of Sn-Xi + . . . + Xn. Assume thatE(X. ) μ and V(X) σ2 for i (1.2. , n}
Please answer question (a) X1 - X X2 – Å a. Let X1, ..., Xn i.i.d....
Please answer question (a)
X1 - X X2 – Å a. Let X1, ..., Xn i.i.d. random variables with X; ~ N(u, o). Express the vector in the | Xn – form AX and find its mean and variance covariance matrix. Show some typical elements of the vari- ance covariance matrix. b. Refer to question (a). The sample variance is given by S2 = n11 21–1(X; – X)2, which can be ex- pressed as S2 = n1X'(I – 111')X (why?)....
Let Xi=I(treatment for ith patient is successful).Then Pr(Xi=1|P=p)=p. Suppose that conditionally...
Let Xi=I(treatment for ith patient is successful).Then Pr(Xi=1|P=p)=p. Suppose that conditionally P=p, X1,X2,...Xn are independent and X=sum of Xi (x from 1 to n). Suppose P~U(0,1) (uniform distribution), want to find the EX. Could you please show me the steps of this question? Thank you so much!
5. Let X1,X2, . , Xn be a random sample from a distribution with finite variance. Show that (i) COV(Xi-X, X )-0 f ) ρ (Xi-XX,-X)--n-1, 1 # J, 1,,-1, , n. OV&.for any two random variables X and Y)...
5. Let X1,X2, . , Xn be a random sample from a distribution with finite variance. Show that (i) COV(Xi-X, X )-0 f ) ρ (Xi-XX,-X)--n-1, 1 # J, 1,,-1, , n. OV&.for any two random variables X and Y) or each 1, and (11 CoV(X,Y) var(x)var(y) (Recall that p vararo
5. Let X1,X2, . , Xn be a random sample from a distribution with finite variance. Show that (i) COV(Xi-X, X )-0 f ) ρ (Xi-XX,-X)--n-1, 1 # J,...
3. Let Xi, . . . , Xn be iid randoln variables with mean μ and...
3. Let Xi, . . . , Xn be iid randoln variables with mean μ and variance σ2. Let, X denote the sample mean and V-Σ, (X,-X)2. (a) Derive the expected values of X and V. (b) Further suppose that Xi,-.,X, are normally distributed. Let Anxn ((a)) an orthogonal matrix whose first rOw 1S be , ..*) and iet Y = AX, where Y (Yİ, ,%), ard X-(XI, , X.), are (column) vectors. (It is not necessary to know aij...
Problem 4 Suppose X1, ..., Xn ~ f(x) independently. Let u = E(Xi) and o2 = Var(Xi). Let X Xi/n. (1) Calculate E(X) and...
Problem 4 Suppose X1, ..., Xn ~ f(x) independently. Let u = E(Xi) and o2 = Var(Xi). Let X Xi/n. (1) Calculate E(X) and Var(X) (2) Explain that X -> u as n -> co. What is the shape of the density of X? (3) Let XiBernoulli(p), calculate u and a2 in terms of p. (4) Continue from (3), explain that X is the frequency of heads. Calculate E(X) and Var(X). Explain that X -> p. What is the shape...
3. Let X1, . . . , Xn be iid random variables with mean μ and...
3. Let X1, . . . , Xn be iid random variables with mean μ and variance σ2. Let X denote the sample mean and V-Σ,(X,-X)2 a) Derive the expected values of X and V b) Further suppose that Xi,...,Xn are normally distributed. Let Anxn - ((a) be an orthogonal matrix whose first row is (mVm Y = (y, . . . ,%), and X = (Xi, , Xn), are (column) vectors. (It is not necessary to know aij for...
- Let {Xn} denote a sequence of iid random variables such that P(Xi = 1) =...
- Let {Xn} denote a sequence of iid random variables such that P(Xi = 1) = P(X1 = -1) = 1/2. Let Sn = X1 + X2 + ... + xn. (a) Find ES, and var(Sn); (b) Show that Sn is a martingale.
Let X1, X2, X3 … be independent random variable with P(Xi = 1) = p =...
Let X1, X2, X3 … be independent random variable with P(Xi = 1) = p = 1-P(Xi=0), i ≥ 1. Define: N1 = min {n: X1+…+ Xn =5}, N2 = 3 if X1 = 0, 5 if X1 = 1. N3 = 3 if X4 = 0, 2 if X4 = 1. Which of the Ni are stopping times for the sequence X1, …?
Let X0,X1,... be a Markov chain whose state space is Z (the integers). Recall the Markov property: P(Xn = in | X0 = i0,X1 = i1,...,Xn−1 = in−1) = P(Xn = in | Xn−1 = in−1), ∀n, ∀it. Does the following...
Let X0,X1,... be a Markov chain whose state space is Z (the
integers).
Recall the Markov property: P(Xn = in | X0 = i0,X1 = i1,...,Xn−1
= in−1) = P(Xn = in | Xn−1 = in−1), ∀n, ∀it. Does the following
always hold: P(Xn ≥0|X0 ≥0,X1 ≥0,...,Xn−1 ≥0)=P(Xn ≥0|Xn−1 ≥0)
?
(Prove if “yes”, provide a counterexample if “no”)
Let Xo,Xi, be a Markov chain whose state space is Z (the integers). Recall the Markov property: P(X,-'n l Xo-io, Xi...
Let Xi, ..., Xn be random variables with the same mean and with covariance function where |ρ| < 1 . Find the mean and variance of Sn-Xi + . . . + Xn. Assume thatE(X. ) μ and V(X) σ2 for i (1.2. , n}
Please answer question (a)
X1 - X X2 – Å a. Let X1, ..., Xn i.i.d. random variables with X; ~ N(u, o). Express the vector in the | Xn – form AX and find its mean and variance covariance matrix. Show some typical elements of the vari- ance covariance matrix. b. Refer to question (a). The sample variance is given by S2 = n11 21–1(X; – X)2, which can be ex- pressed as S2 = n1X'(I – 111')X (why?)....
5. Let X1,X2, . , Xn be a random sample from a distribution with finite variance. Show that (i) COV(Xi-X, X )-0 f ) ρ (Xi-XX,-X)--n-1, 1 # J, 1,,-1, , n. OV&.for any two random variables X and Y) or each 1, and (11 CoV(X,Y) var(x)var(y) (Recall that p vararo
5. Let X1,X2, . , Xn be a random sample from a distribution with finite variance. Show that (i) COV(Xi-X, X )-0 f ) ρ (Xi-XX,-X)--n-1, 1 # J,...
3. Let Xi, . . . , Xn be iid randoln variables with mean μ and variance σ2. Let, X denote the sample mean and V-Σ, (X,-X)2. (a) Derive the expected values of X and V. (b) Further suppose that Xi,-.,X, are normally distributed. Let Anxn ((a)) an orthogonal matrix whose first rOw 1S be , ..*) and iet Y = AX, where Y (Yİ, ,%), ard X-(XI, , X.), are (column) vectors. (It is not necessary to know aij...
Problem 4 Suppose X1, ..., Xn ~ f(x) independently. Let u = E(Xi) and o2 = Var(Xi). Let X Xi/n. (1) Calculate E(X) and Var(X) (2) Explain that X -> u as n -> co. What is the shape of the density of X? (3) Let XiBernoulli(p), calculate u and a2 in terms of p. (4) Continue from (3), explain that X is the frequency of heads. Calculate E(X) and Var(X). Explain that X -> p. What is the shape...
3. Let X1, . . . , Xn be iid random variables with mean μ and variance σ2. Let X denote the sample mean and V-Σ,(X,-X)2 a) Derive the expected values of X and V b) Further suppose that Xi,...,Xn are normally distributed. Let Anxn - ((a) be an orthogonal matrix whose first row is (mVm Y = (y, . . . ,%), and X = (Xi, , Xn), are (column) vectors. (It is not necessary to know aij for...
- Let {Xn} denote a sequence of iid random variables such that P(Xi = 1) = P(X1 = -1) = 1/2. Let Sn = X1 + X2 + ... + xn. (a) Find ES, and var(Sn); (b) Show that Sn is a martingale.
Let X0,X1,... be a Markov chain whose state space is Z (the
integers).
Recall the Markov property: P(Xn = in | X0 = i0,X1 = i1,...,Xn−1
= in−1) = P(Xn = in | Xn−1 = in−1), ∀n, ∀it. Does the following
always hold: P(Xn ≥0|X0 ≥0,X1 ≥0,...,Xn−1 ≥0)=P(Xn ≥0|Xn−1 ≥0)
?
(Prove if “yes”, provide a counterexample if “no”)
Let Xo,Xi, be a Markov chain whose state space is Z (the integers). Recall the Markov property: P(X,-'n l Xo-io, Xi...
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