Only need parts c, e, j, m, and p
only need parts c, e, j, m, and p
Homework Answers
help me find point a-p are independent. distribution or otherwise state "unknown." that X, ~, ΝΟμ,...
help me find point a-p
are independent. distribution or otherwise state "unknown." that X, ~, ΝΟμ, σ 2), l a. 1, '.. , n and Z, ,~ NO. 1), i l' k and all variables State the distribution of each of the following variables if'it is a "named" (b) X, + 2X) nk(X-μ) (d) zi nx - 1) (e)S (k-1) Σ(Xi-X) Y NU6 251 and
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,...
EXERCISE 6 Let Zi, Z2,-.., Zi6 be an i.i.d. sample of size 16 from the standard...
EXERCISE 6 Let Zi, Z2,-.., Zi6 be an i.i.d. sample of size 16 from the standard normal distribution N (0,1). Let Xi,X2,..., X64 be an i.i.d. sample of size 64 from the normal distribution (μ, 1). The two samples are independent. a. What is the distribution of Y, where Y-Σ161 Z2 + Σ-i(X-μ)2? Study List b. Find ΕΥ. c. Find Var(Y) d. Approximate P(Y105)
9. Consider the following hidden Markov model (HMM) (This is the same HMM as in the previous HMM problem): ·X=(x, ,...
9. Consider the following hidden Markov model (HMM) (This is the same HMM as in the previous HMM problem): ·X=(x, ,x,Je {0,1)、[i.e., X is a binary sequence of length n] and Y-(Y Rt [i.e. Y is a sequence of n real numbers.) ·X1~" Bernoulli(1/2) ,%) E Ip is the switching probability; when p is small the Markov chain likes to stay in the same state] . conditioned on X, the random variables Yı , . . . , y, are...
2. The random variables X1, X2 and X3 are independent, with Xi N(0,1), X2 N(1,4) and...
2. The random variables X1, X2 and X3 are independent, with Xi N(0,1), X2 N(1,4) and X3 ~ N(-1.2). Consider the random column vector X-Xi, X2,X3]T. (a) Write X in the form where Z is a vector of iid standard normal random variables, μ is a 3x vector, and B is a 3 × 3 matrix. (b) What is the covariance matrix of X? (c) Determine the expectation of Yi = Xi + X3. (d) Determine the distribution of Y2...
Rst of Robert V. Hogg, Allen Craig-lntroduction t no say e instances it ma ul roblem ean be avoid...
proof the theorem 1
rst of Robert V. Hogg, Allen Craig-lntroduction t no say e instances it ma ul roblem ean be avoided if we will but prove the following factorization theorem of Neyman. Theorem 1. Let a distribution that has pdf. f(x, θ), θ62. The statistic Y1= n(x, , x2, . . . , X") is a suficient statistic for θ if and only if we can find two nonnegative functions, kj and k2, such that t X,, X2,...,...
4. Suppose X1, . . . ,X, are independent, normally distributed with mean E(Xi) and variance...
4. Suppose X1, . . . ,X, are independent, normally distributed with mean E(Xi) and variance Var(X)-σί. Let Żi-(X,-μ.)/oi so that Zi , . . . , Ζ,, are independent and each has a N(0, 1) distribution. Show that LZhas a x2 distribution. Hint: Use the fact that each Z has a xî distribution i naS
need to check my work. Just need B and C Problem 2. Recall that a discrete...
need to check my work. Just
need B and C
Problem 2. Recall that a discrete random variable X has Poisson distribution with parameter λ if the probability mass function of X is fx (x) = e-λ- XE(0, 1,2, ) ar! This distribution is often used to model the number of events which will occur in a given time span, given that λ such events occur on average a Prove by direct cornputation that the mean of a Poisson randoln...
e (4 marks) Let m be an integer with the property that m 2 2. Consider that X1, X2,.. ., Xm are independent Binomial(n,p) random variables, where n is known and p is unknown. Note that p E (0,1). Wr...
e (4 marks) Let m be an integer with the property that m 2 2. Consider that X1, X2,.. ., Xm are independent Binomial(n,p) random variables, where n is known and p is unknown. Note that p E (0,1). Write down the expression of the likelihood function We assume that min(x1, . . . ,xm) 〈 n and max(x1, . . . ,xm) 〉 0 5 marks) Find , and give all possible solutions to the equation dL dL -...
Central Limit Theorem: let x1,x2,...,xn be I.I.D. random variables with E(xi)= U Var(xi)= (sigma)^2 defind Z=...
Central Limit Theorem: let x1,x2,...,xn be I.I.D. random variables with E(xi)= U Var(xi)= (sigma)^2 defind Z= x1+x2+...+xn the distribution of Z converges to a gaussian distribution P(Z<=z)=1-Q((z-Uz)/(sigma)^2) Use MATLAB to prove the central limit theorem. To achieve this, you will need to generate N random variables (I.I.D. with the distribution of your choice) and show that the distribution of the sum approaches a Guassian distribution. Plot the distribution and matlab code. Hint: you may find the hist() function helpful
help me find point a-p
are independent. distribution or otherwise state "unknown." that X, ~, ΝΟμ, σ 2), l a. 1, '.. , n and Z, ,~ NO. 1), i l' k and all variables State the distribution of each of the following variables if'it is a "named" (b) X, + 2X) nk(X-μ) (d) zi nx - 1) (e)S (k-1) Σ(Xi-X) Y NU6 251 and
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,...
EXERCISE 6 Let Zi, Z2,-.., Zi6 be an i.i.d. sample of size 16 from the standard normal distribution N (0,1). Let Xi,X2,..., X64 be an i.i.d. sample of size 64 from the normal distribution (μ, 1). The two samples are independent. a. What is the distribution of Y, where Y-Σ161 Z2 + Σ-i(X-μ)2? Study List b. Find ΕΥ. c. Find Var(Y) d. Approximate P(Y105)
9. Consider the following hidden Markov model (HMM) (This is the same HMM as in the previous HMM problem): ·X=(x, ,x,Je {0,1)、[i.e., X is a binary sequence of length n] and Y-(Y Rt [i.e. Y is a sequence of n real numbers.) ·X1~" Bernoulli(1/2) ,%) E Ip is the switching probability; when p is small the Markov chain likes to stay in the same state] . conditioned on X, the random variables Yı , . . . , y, are...
2. The random variables X1, X2 and X3 are independent, with Xi N(0,1), X2 N(1,4) and X3 ~ N(-1.2). Consider the random column vector X-Xi, X2,X3]T. (a) Write X in the form where Z is a vector of iid standard normal random variables, μ is a 3x vector, and B is a 3 × 3 matrix. (b) What is the covariance matrix of X? (c) Determine the expectation of Yi = Xi + X3. (d) Determine the distribution of Y2...
proof the theorem 1
rst of Robert V. Hogg, Allen Craig-lntroduction t no say e instances it ma ul roblem ean be avoided if we will but prove the following factorization theorem of Neyman. Theorem 1. Let a distribution that has pdf. f(x, θ), θ62. The statistic Y1= n(x, , x2, . . . , X") is a suficient statistic for θ if and only if we can find two nonnegative functions, kj and k2, such that t X,, X2,...,...
4. Suppose X1, . . . ,X, are independent, normally distributed with mean E(Xi) and variance Var(X)-σί. Let Żi-(X,-μ.)/oi so that Zi , . . . , Ζ,, are independent and each has a N(0, 1) distribution. Show that LZhas a x2 distribution. Hint: Use the fact that each Z has a xî distribution i naS
need to check my work. Just
need B and C
Problem 2. Recall that a discrete random variable X has Poisson distribution with parameter λ if the probability mass function of X is fx (x) = e-λ- XE(0, 1,2, ) ar! This distribution is often used to model the number of events which will occur in a given time span, given that λ such events occur on average a Prove by direct cornputation that the mean of a Poisson randoln...
e (4 marks) Let m be an integer with the property that m 2 2. Consider that X1, X2,.. ., Xm are independent Binomial(n,p) random variables, where n is known and p is unknown. Note that p E (0,1). Write down the expression of the likelihood function We assume that min(x1, . . . ,xm) 〈 n and max(x1, . . . ,xm) 〉 0 5 marks) Find , and give all possible solutions to the equation dL dL -...
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