Question

In: Statistics and Probability

1. Consider the following weighted averages of independent random variables X1, X2, X3, all with mean...

1. Consider the following weighted averages of independent random variables X1, X2, X3, all with mean u and variance σ^2

θ1 = 1/3(X1) + 1/3(X2) + 1/3(X3)

θ2 = 1/4(X1) + 2/4(X2) + 1/4(X3)

θ3 = 2/5(X1) + 2/5(X2) + 2/5(X3)

a) Find E[θ1], E[θ2], E[θ3]

b) Are θ1, θ2 and θ3 unbiased for u? Explain

c) Find the variance for θ1, θ2 and θ3

d) If you had to use one of the above estimators, which would you pick? Explain

Expert Solution

a)

b)

.

c) We will use the fact that are independent here.

d) We would like to choose an unbiased estimator with the least variance.

The estimator has variance , while has variance

Since, the variance for is lesser than that of we will prefer the estimator . You can also note that the variance of is greater than , and we always want an estimator with the least variance.

orchestra answered 3 years ago

Consider independent random variables X1, X2, and X3 such that X1 is a random variable having...

Consider independent random variables X1, X2, and X3 such that X1 is a random variable having mean 1 and variance 1, X2 is a random variable having mean 2 and variance 4, and X3 is a random variable having mean 3 and variance 9. (a) Give the value of the variance of X1 + (1/2)X2 + (1/3)X3 (b) Give the value of the correlation of Y = X1- X2 and Z = X2 + X3.

Consider a sequence of random variables X0, X1, X2, X3, . . . which form a...

Consider a sequence of random variables X0, X1, X2, X3, . . . which form a Markov chain. (a) Define the Markov property for this Markov chain both in words and using a mathematical formula. (b) When is a Markov chain irreducible? (c) Give the definition for an ergodic state.

Let X1, X2, X3 be continuous random variables with joint pdf f(X1, X2, X3)= 2 if...

Let X1, X2, X3 be continuous random variables with joint pdf f(X1, X2, X3)= 2 if 1<X1<2 -1<X2<0 -X2-1<X3<0 0 otherwise Find Cov(X2, X3)

Let X1,X2,X3 be i.i.d. N(0,1) random variables. Suppose Y1 = X1 + X2 + X3, Y2...

Let X1,X2,X3 be i.i.d. N(0,1) random variables. Suppose Y1 = X1 + X2 + X3, Y2 = X1 −X2, Y3 =X1 −X3. Find the joint pdf of Y = (Y1,Y2,Y3)′ using : Multivariate normal distribution properties.

let X1, X2, X3 be random variables that are defined as X1 = θ + ε1...

let X1, X2, X3 be random variables that are defined as X1 = θ + ε1 X2 = 2θ + ε2 X3 = 3θ + ε3 ε1, ε2, ε3 are independent and the mean and variance are the following random variable E(ε1) = E(ε2) = E(ε3) = 0 Var(ε1) = 4 Var(ε2) = 6 Var(ε3) = 8 What is the Best Linear Unbiased Estimator(BLUE) when estimating parameter θ from the three samples X1, X2, X3

Suppose that X1,X2,X3,X4 are independent random variables with common mean E(Xi) =μ and variance Var(Xi) =σ2....

Suppose that X1,X2,X3,X4 are independent random variables with common mean E(Xi) =μ and variance Var(Xi) =σ2. LetV=X2−X3+X4 and W=X1−2X2+X3+ 4X4. (a) Find E(V) and E(W). (b) Find Var(V) and Var(W). (c) Find Cov(V,W).( d) Find the correlation coefficientρ(V,W). Are V and W independent?

Suppose that X1,X2,X3,X4 are independent random variables with common mean E(Xi) =μ and variance Var(Xi) =σ2....

Suppose that X1,X2,X3,X4 are independent random variables with common mean E(Xi) =μ and variance Var(Xi) =σ2. LetV=X2−X3+X4 and W=X1−2X2+X3+ 4X4. (a) Find E(V) and E(W). (b) Find Var(V) and Var(W). (c) Find Cov(V,W).( d) Find the correlation coefficientρ(V,W). Are V and W independent?

Let X1, X2, X3, X4, X5 be independent continuous random variables having a common cdf F...

Let X1, X2, X3, X4, X5 be independent continuous random variables having a common cdf F and pdf f, and set p=P(X1 <X2 <X3 < X4 < X5). (i) Show that p does not depend on F. Hint: Write I as a five-dimensional integral and make the change of variables ui = F(xi), i = 1,··· ,5. (ii) Evaluate p. (iii) Give an intuitive explanation for your answer to (ii).

X1 and X2 are two independent random variables that have Poisson distributions with mean lambda1 and...

X1 and X2 are two independent random variables that have Poisson distributions with mean lambda1 and lambda2, respectively. a) Use moment generating functions, derive and name the distribution of X = X1+X2 b) Derive and name the conditional distribution of X1 given that X = N where N is a fixed positive integer. Please explain your answer in detail. Please don't copy other answers. Thank you; will thumb up!

Consider random variables X1, X2, and X3 with binomial distribution, uniform, and normal probability density functions...

Consider random variables X1, X2, and X3 with binomial distribution, uniform, and normal probability density functions (PDF) respectively. Generate list of 50 random values, between 0 and 50, for these variables and store them with names Data1, Data2, and Data3 respectively. ( To complete successfully this homework on Stochastic Models, you need to use one of the software tools: Excel, SPSS or Mathematica, to answer the following items, and print out your results directly from the software.)

Question

1. Consider the following weighted averages of independent random variables X1, X2, X3, all with mean...

Solutions

Expert Solution

Related Solutions

Consider independent random variables X1, X2, and X3 such that X1 is a random variable having...

Consider a sequence of random variables X0, X1, X2, X3, . . . which form a...

Let X1, X2, X3 be continuous random variables with joint pdf f(X1, X2, X3)= 2 if...

Let X1,X2,X3 be i.i.d. N(0,1) random variables. Suppose Y1 = X1 + X2 + X3, Y2...

let X1, X2, X3 be random variables that are defined as X1 = θ + ε1...

Suppose that X1,X2,X3,X4 are independent random variables with common mean E(Xi) =μ and variance Var(Xi) =σ2....

Suppose that X1,X2,X3,X4 are independent random variables with common mean E(Xi) =μ and variance Var(Xi) =σ2....

Let X1, X2, X3, X4, X5 be independent continuous random variables having a common cdf F...

X1 and X2 are two independent random variables that have Poisson distributions with mean lambda1 and...

Consider random variables X1, X2, and X3 with binomial distribution, uniform, and normal probability density functions...