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2. The random variables X1, X2 and X3 are independent, with Xi N(0,1), X2 N(1,4) and...
7. Let X1 and X2 be two iid exp(A) random variables. Set Yi Xi - X2...
7. Let X1 and X2 be two iid exp(A) random variables. Set Yi Xi - X2 and Y2 X + X2. Determine the joint pdf of Y and Y2, identify the marginal distributions of Yi and Y2, and decide whether or not Yi and Y2 are independent [10)
If X1, X2, and X3 are three independent Uniform random variables (Xi-Unif(0,1)) a) Use the convol...
If X1, X2, and X3 are three independent Uniform random variables (Xi-Unif(0,1)) a) Use the convolution integral to find density function of Z-x1+X2+X3. b) What is E[Z]? independent Uniform random variables (Xi-Unifo,1):
If X1, X2, and X3 are three independent Uniform random variables (Xi-Unif(0,1)) a) Use the convolution integral to find density function of Z-x1+X2+X3. b) What is E[Z]? independent Uniform random variables (Xi-Unifo,1):
3. (25 pts.) Let X1, X2, X3 be independent random variables such that Xi~ Poisson (A), i 1,2,3. Let N = X1 + X2+X3. (a)...
3. (25 pts.) Let X1, X2, X3 be independent random variables such that Xi~ Poisson (A), i 1,2,3. Let N = X1 + X2+X3. (a) What is the distribution of N? (b) Find the conditional distribution of (X1, X2, X3) | N. (c) Now let N, X1, X2, X3, be random variables such that N~ Poisson(A), (X1, X2, X3) | N Trinomial(N; pi,p2.ps) where pi+p2+p3 = 1. Find the unconditional distribution of (X1, X2, X3).
3. (25 pts.) Let X1,...
Probabilities Compute P(X1 + 2X2 -2X3 > 7) , where X1, X2, X3 are iid random variables with common distribution N(1,4). You must give the numerical answer. Specify and show any formulas used for E,...
Probabilities Compute P(X1 + 2X2 -2X3 > 7) , where X1, X2, X3 are iid random variables with common distribution N(1,4). You must give the numerical answer. Specify and show any formulas used for E, P, and Var for example Var( X1 + 2X2 -2X3) = 4 + 22(4) + (-2)2(4) = 35. Please specify how all the individual numbers where obtained and the formula used.
Let X1,X2,X3..Xn be iid of f(x)= theta. x^(theta-1), with x(0,1) and theta being a positive numbe...
Let X1,X2,X3..Xn be iid of f(x)= theta. x^(theta-1), with x(0,1) and theta being a positive number. Is the parameter identifiable?.Compute the maximum likelihood estimate. If instead of X1,X2,,, We observe, Y1,Y2,...Yn, where Yi=1(Xi<=0.5).What distribution does Yi follow? What is the parameter of this distribution? Compute MLE and the method of moments and Fisher information.
Let X1,X2,X3..Xn be iid of f(x)= theta. x^(theta-1), with x(0,1) and theta being a positive number....
Let X1,X2,X3..Xn be iid of f(x)= theta. x^(theta-1), with x(0,1) and theta being a positive number. Is the parameter identifiable?.Compute the maximum likelihood estimate. If instead of X1,X2,,, We observe, Y1,Y2,...Yn, where Yi=1(Xi<=0.5).What distribution does Yi follow? What is the parameter of this distribution? Compute MLE and the method of moments and Fisher information.
5. Suppose that three random variables Xi, X2, and X3 have a continuous joint distribution with...
5. Suppose that three random variables Xi, X2, and X3 have a continuous joint distribution with the following p.d.f. (x1+2x2+3z3) and f(1, r2, 3) 0 otherwise. (a) Determine the value of the constant c; (b) Find the marginal joint p.d.f. of Xi and X3; (c) Find P(Xi < 1|X2-2, X3-1)
Let X1, X2, and X3 be uncorrelated random variables, each with 4. (10 points) Let Xi,...
Let X1, X2, and X3 be uncorrelated random variables, each
with
4. (10 points) Let Xi, X2, and X3 be uncorrelated random variables, each with mean u and variance o2. Find, in terms of u and o2 a) Cov(X+ 2X2, X7t 3X;) b) Cov(Xrt X2, Xi- X2)
4.) Let X1, X2 and X3 be independent uniform random variables on [0,1]. Write Y =...
4.) Let X1, X2 and X3 be independent uniform random variables on [0,1]. Write Y = X1 + X, and Z X2 + X3 a.) Compute E[X, X,X3]. (5 points) b.) Compute Var(x1). (5 points) c.) Compute and draw a graph of the density function fy (15 points)
Q2 Suppose X1, X2, X3 are independent Bernoulli random variables with p = 0.5. Let Y;...
Q2 Suppose X1, X2, X3 are independent Bernoulli random variables with p = 0.5. Let Y; be the partial sums, i.e., Y1 = X1, Y2 = X1 + X2, Y3 = X1 + X2 + X3. 1. What is the distubution for each Yį, i = 1, 2, 3? 2. What is the expected value for Y1 + Y2 +Yz? 3. Are Yį and Y2 independent? Explain it by computing their joint P.M.F. 4. What is the variance of Y1...
7. Let X1 and X2 be two iid exp(A) random variables. Set Yi Xi - X2 and Y2 X + X2. Determine the joint pdf of Y and Y2, identify the marginal distributions of Yi and Y2, and decide whether or not Yi and Y2 are independent [10)
If X1, X2, and X3 are three independent Uniform random variables (Xi-Unif(0,1)) a) Use the convolution integral to find density function of Z-x1+X2+X3. b) What is E[Z]? independent Uniform random variables (Xi-Unifo,1):
If X1, X2, and X3 are three independent Uniform random variables (Xi-Unif(0,1)) a) Use the convolution integral to find density function of Z-x1+X2+X3. b) What is E[Z]? independent Uniform random variables (Xi-Unifo,1):
3. (25 pts.) Let X1, X2, X3 be independent random variables such that Xi~ Poisson (A), i 1,2,3. Let N = X1 + X2+X3. (a) What is the distribution of N? (b) Find the conditional distribution of (X1, X2, X3) | N. (c) Now let N, X1, X2, X3, be random variables such that N~ Poisson(A), (X1, X2, X3) | N Trinomial(N; pi,p2.ps) where pi+p2+p3 = 1. Find the unconditional distribution of (X1, X2, X3).
3. (25 pts.) Let X1,...
5. Suppose that three random variables Xi, X2, and X3 have a continuous joint distribution with the following p.d.f. (x1+2x2+3z3) and f(1, r2, 3) 0 otherwise. (a) Determine the value of the constant c; (b) Find the marginal joint p.d.f. of Xi and X3; (c) Find P(Xi < 1|X2-2, X3-1)
Let X1, X2, and X3 be uncorrelated random variables, each
with
4. (10 points) Let Xi, X2, and X3 be uncorrelated random variables, each with mean u and variance o2. Find, in terms of u and o2 a) Cov(X+ 2X2, X7t 3X;) b) Cov(Xrt X2, Xi- X2)
4.) Let X1, X2 and X3 be independent uniform random variables on [0,1]. Write Y = X1 + X, and Z X2 + X3 a.) Compute E[X, X,X3]. (5 points) b.) Compute Var(x1). (5 points) c.) Compute and draw a graph of the density function fy (15 points)
Q2 Suppose X1, X2, X3 are independent Bernoulli random variables with p = 0.5. Let Y; be the partial sums, i.e., Y1 = X1, Y2 = X1 + X2, Y3 = X1 + X2 + X3. 1. What is the distubution for each Yį, i = 1, 2, 3? 2. What is the expected value for Y1 + Y2 +Yz? 3. Are Yį and Y2 independent? Explain it by computing their joint P.M.F. 4. What is the variance of Y1...
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