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Consider k groups and considerYi,Y2...,Yk independent random variables, where Yi ~b(ni,P)1,2,.., k, Let ni be a...
Problem 2. (5 marks. 3, 2) Let Yi and Y2 be two independent discrete random variables...
Problem 2. (5 marks. 3, 2) Let Yi and Y2 be two independent discrete random variables such that: pi (yi) = ,--2-1, 0 and P2(U2) = 2 = 1.6 Let K = Yi + Y2. a) Find the moment generating function of Y1,Y2 and K. b) Using part a), find the probability mass function of K
Exercise 6 Let Yi, Y2, Ys be independent random variables with distribution N (i, i2) for...
Exercise 6 Let Yi, Y2, Ys be independent random variables with distribution N (i, i2) for i = 1, 2, 3 (that is, each is normally distributed with mean mean E(Y) = i and variance V(X) = i2). For each of the following situations, use the Y, i = 1, 2, 3 to construct a statistic with the indicated distribution a) X2 with 3 degrees of freedom b) t distribution with 2 degrees of freedom c) F distribution with 1...
3. Let yi and ya have the joint density function otherwise, the same as in the previous problem. a) Show that yi and Y2 are dependent random variables. b) Note that when the joint density can be w...
3. Let yi and ya have the joint density function otherwise, the same as in the previous problem. a) Show that yi and Y2 are dependent random variables. b) Note that when the joint density can be written as the product of a function of n and a function of 32 - which is the case here- the 2 random variables would be independent if the joint density is nonzero on a rectangular domain, according to a theorem we learned....
8.7-11. Let Y1,Y2, ...,Yn be n independent random variables with normal distributions N(Bx;,02), where X],x2,...,xn are...
8.7-11. Let Y1,Y2, ...,Yn be n independent random variables with normal distributions N(Bx;,02), where X],x2,...,xn are known and not all equal and B and 2 are unknown parameters (a) Find the likelihood ratio test for Ho: B = 0 against H: B+0. (b) Can this test be based on a statistic with a well-known distribution?
4. (24 marks) Suppose that the random variables Yi,..., Yn satisfy Y-B BX,+ Ei, 1-1, , n, where βο and βι are parameters, X1, ,X, are con- stants, and e1,... ,en are independent and identically distr...
4. (24 marks) Suppose that the random variables Yi,..., Yn satisfy Y-B BX,+ Ei, 1-1, , n, where βο and βι are parameters, X1, ,X, are con- stants, and e1,... ,en are independent and identically distributed ran- dom variables with Ei ~ N (0,02), where σ2 is a third unknown pa- rameter. This is the familiar form for a simple linear regression model, where the parameters A, β, and σ2 explain the relationship between a dependent (or response) variable Y...
Problem 2. (5 marks. 3, 2) Let Yi and Y2 be two independent discrete random variables such that: pi (yi) = ,--2-1, 0 and P2(U2) = 2 = 1.6 Let K = Yi + Y2. a) Find the moment generating function of Y1,Y2 and K. b) Using part a), find the probability mass function of K
Exercise 6 Let Yi, Y2, Ys be independent random variables with distribution N (i, i2) for i = 1, 2, 3 (that is, each is normally distributed with mean mean E(Y) = i and variance V(X) = i2). For each of the following situations, use the Y, i = 1, 2, 3 to construct a statistic with the indicated distribution a) X2 with 3 degrees of freedom b) t distribution with 2 degrees of freedom c) F distribution with 1...
3. Let yi and ya have the joint density function otherwise, the same as in the previous problem. a) Show that yi and Y2 are dependent random variables. b) Note that when the joint density can be written as the product of a function of n and a function of 32 - which is the case here- the 2 random variables would be independent if the joint density is nonzero on a rectangular domain, according to a theorem we learned....
8.7-11. Let Y1,Y2, ...,Yn be n independent random variables with normal distributions N(Bx;,02), where X],x2,...,xn are known and not all equal and B and 2 are unknown parameters (a) Find the likelihood ratio test for Ho: B = 0 against H: B+0. (b) Can this test be based on a statistic with a well-known distribution?
4. (24 marks) Suppose that the random variables Yi,..., Yn satisfy Y-B BX,+ Ei, 1-1, , n, where βο and βι are parameters, X1, ,X, are con- stants, and e1,... ,en are independent and identically distributed ran- dom variables with Ei ~ N (0,02), where σ2 is a third unknown pa- rameter. This is the familiar form for a simple linear regression model, where the parameters A, β, and σ2 explain the relationship between a dependent (or response) variable Y...
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