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Suppose X ~ N(M, M2), but instead of assuming u is a constant, we now make...
Question 3 (5101) Suppose that N ~ Poisson(2), and that X has a conditional distribution that...
Question 3 (5101) Suppose that N ~ Poisson(2), and that X has a conditional distribution that depends on N. There are two possible models for this conditional distribution: • Model M: X|N=n~ N(2*n, 02 = 1) • Model M2: X|N=n~ N(2+n + n2,02 = 1) The only difference between M, and M2 is the conditional mean function - the mean function is linear in M, and quadratic in M2. a If M is true: Find E(X) and V. [X] as...
Suppose that U is a random variable with a uniform distribution on (0,1). Now suppose that...
Suppose that U is a random variable with a uniform distribution on (0,1). Now suppose that f is the PDF of some continuous random variable of interest, that F is the corresponding CDF, and assume that F is invertible (so that the function F-1 exists and gives a unique value). Show that the random variable X = F-1(U) has PDF f(x)—that is, that X has the desired PDF. Hint: use results on transformations of random variables. This cute result allows...
Question 3 (5101) Suppose that N ~ Poisson(A), and that X has a conditional distribution that...
Question 3 (5101) Suppose that N ~ Poisson(A), and that X has a conditional distribution that depends on N. There are two possible models for this conditional distribution: . Model M: X|N=n~ N(2*n, o2 = 1) • Model M2: X|N=n~ N(2+n + n2,02 = 1) The only difference between M and M2 is the conditional mean function - the mean function is linear in M, and quadratic in M2. a If Mi is true: Find Ej[X] and V1 [X] as...
6. The median, or Q2, is nothing special. When creating a sign test, we could have designed the testing infrastructure...
6. The median, or Q2, is nothing special. When creating a sign test, we could have designed the testing infrastructure around the first quartile, Q1, instead. Suppose we set a null hypothesis of Ho: q = go, where q is a symbol for the first quartile, go is a constant, and we are dealing with a continuous random variable X. a. Assuming Ho, we collect n values from X: Xi,..., Xn If U is the number of observations less than...
1. Solve Utt - 4lxx- u(x,0)-ar m(x,0) = 0. b. Now we replace the initial condition in a with u(x,...
1. Solve Utt - 4lxx- u(x,0)-ar m(x,0) = 0. b. Now we replace the initial condition in a with u(x,0)-x 1 x e [o,1 0,1 Let ua be the solution of a and let u be the solution of b. Find the set of all x E R such that 14(x, 5) = 11 b(x,5).
1. Solve Utt - 4lxx- u(x,0)-ar m(x,0) = 0. b. Now we replace the initial condition in a with u(x,0)-x 1 x e [o,1 0,1...
1. Suppose that Y ∼ Gamma(α, β) and c > 0 is a constant. (a) Derive the density function of U ...
1. Suppose that Y ∼ Gamma(α, β) and c > 0 is a constant. (a)
Derive the density function of U = cY. (b) Identify the
distribution of U as a standard distribution. Be sure to identify
any parameter values. (c) Can you find the distribution of U using
MGF method also?
I. Suppose that Y ~ Gamma(α, β) and c > 0 is a constant. (a) Derive the density function of U cY. (b) Identify the distribution of U...
Suppose X and Y are independent and Prove the following a) U=X+Y~gamma(α + β,γ) b) V=X/(X...
Suppose X and Y are independent and
Prove the following
a) U=X+Y~gamma(α + β,γ)
b) V=X/(X + Y ) ∼ beta(α,β)
c) U, V independent
d) ~gamma(1/2,
1/2) when W~N(0,1)
X ~ gammala, y) and Y ~ gamma(6, 7) We were unable to transcribe this image
Having troubles with question 2. Please help 2. If X has a Gamma distribution with parameters...
Having troubles with question 2. Please help
2. If X has a Gamma distribution with parameters a and B, then its mgf is given by (a) Obtain expressions for the moment-genérating functions of an exponential random variable and of a chi-square random variable by recognizing that these are special cases of a Gamma distribution and using the mgf given above. (b) Suppose that X1 is a Gamma variable with parameters α1 and β, X2 is a Gamma variable with parameters...
7.97 Suppose you want to test Ho: u = 500 against He: p > 500 using...
7.97 Suppose you want to test Ho: u = 500 against He: p > 500 using a = .05. The population in question is normally dis- tributed with standard deviation 100. A random sample of size n = 25 will be used. a. Sketch the sampling distribution of x assuming that Ho is true. b. Find the value of xo, that value of x above which the null hypothesis will be rejected. Indicate the rejection region on your graph of...
Suppose that N ~ Poisson(1), and that X has a conditional distribution that depends on N....
Suppose that N ~ Poisson(1), and that X has a conditional distribution that depends on N. There are two possible models for this conditional distribution: • Model M1: X|N = n ~ N(2* n,o2 = 1) • Model M2: X|N = n ~ N(2*n+n2,02 = 1) The only difference between My and M2 is the conditional mean function - the mean function is linear in M1 and quadratic in M2. a If M1 is true: Find E1[X] and V1[X] as...
Question 3 (5101) Suppose that N ~ Poisson(2), and that X has a conditional distribution that depends on N. There are two possible models for this conditional distribution: • Model M: X|N=n~ N(2*n, 02 = 1) • Model M2: X|N=n~ N(2+n + n2,02 = 1) The only difference between M, and M2 is the conditional mean function - the mean function is linear in M, and quadratic in M2. a If M is true: Find E(X) and V. [X] as...
Suppose that U is a random variable with a uniform distribution on (0,1). Now suppose that f is the PDF of some continuous random variable of interest, that F is the corresponding CDF, and assume that F is invertible (so that the function F-1 exists and gives a unique value). Show that the random variable X = F-1(U) has PDF f(x)—that is, that X has the desired PDF. Hint: use results on transformations of random variables. This cute result allows...
Question 3 (5101) Suppose that N ~ Poisson(A), and that X has a conditional distribution that depends on N. There are two possible models for this conditional distribution: . Model M: X|N=n~ N(2*n, o2 = 1) • Model M2: X|N=n~ N(2+n + n2,02 = 1) The only difference between M and M2 is the conditional mean function - the mean function is linear in M, and quadratic in M2. a If Mi is true: Find Ej[X] and V1 [X] as...
6. The median, or Q2, is nothing special. When creating a sign test, we could have designed the testing infrastructure around the first quartile, Q1, instead. Suppose we set a null hypothesis of Ho: q = go, where q is a symbol for the first quartile, go is a constant, and we are dealing with a continuous random variable X. a. Assuming Ho, we collect n values from X: Xi,..., Xn If U is the number of observations less than...
1. Solve Utt - 4lxx- u(x,0)-ar m(x,0) = 0. b. Now we replace the initial condition in a with u(x,0)-x 1 x e [o,1 0,1 Let ua be the solution of a and let u be the solution of b. Find the set of all x E R such that 14(x, 5) = 11 b(x,5).
1. Solve Utt - 4lxx- u(x,0)-ar m(x,0) = 0. b. Now we replace the initial condition in a with u(x,0)-x 1 x e [o,1 0,1...
1. Suppose that Y ∼ Gamma(α, β) and c > 0 is a constant. (a)
Derive the density function of U = cY. (b) Identify the
distribution of U as a standard distribution. Be sure to identify
any parameter values. (c) Can you find the distribution of U using
MGF method also?
I. Suppose that Y ~ Gamma(α, β) and c > 0 is a constant. (a) Derive the density function of U cY. (b) Identify the distribution of U...
Suppose X and Y are independent and
Prove the following
a) U=X+Y~gamma(α + β,γ)
b) V=X/(X + Y ) ∼ beta(α,β)
c) U, V independent
d) ~gamma(1/2,
1/2) when W~N(0,1)
X ~ gammala, y) and Y ~ gamma(6, 7) We were unable to transcribe this image
Having troubles with question 2. Please help
2. If X has a Gamma distribution with parameters a and B, then its mgf is given by (a) Obtain expressions for the moment-genérating functions of an exponential random variable and of a chi-square random variable by recognizing that these are special cases of a Gamma distribution and using the mgf given above. (b) Suppose that X1 is a Gamma variable with parameters α1 and β, X2 is a Gamma variable with parameters...
7.97 Suppose you want to test Ho: u = 500 against He: p > 500 using a = .05. The population in question is normally dis- tributed with standard deviation 100. A random sample of size n = 25 will be used. a. Sketch the sampling distribution of x assuming that Ho is true. b. Find the value of xo, that value of x above which the null hypothesis will be rejected. Indicate the rejection region on your graph of...
Suppose that N ~ Poisson(1), and that X has a conditional distribution that depends on N. There are two possible models for this conditional distribution: • Model M1: X|N = n ~ N(2* n,o2 = 1) • Model M2: X|N = n ~ N(2*n+n2,02 = 1) The only difference between My and M2 is the conditional mean function - the mean function is linear in M1 and quadratic in M2. a If M1 is true: Find E1[X] and V1[X] as...
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