5. , X, be a random sample from a gamma distribution, X, ~GAM(0, 2),Show Let X1, that x, is sufficient for θ (a) by using equation (10.2.1), (b) by the factorization criterion of equation (10.2.3)
oren. 0.2.1 Factorization Criterion If X1, , X" have joint pdff(x,, , x.. θ), and is (S S then S..., S are jointly sufficient for 0 if and only ift 15 *9 (10.2.3) , Xn) where g(s; θ) does not depend on xi, does not involve 0. , x, , except through s, and h(x,,
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Definition 10.2.3 If A is a set, then the indicator function of A, denoted by I, is defined as 1 ...
The indicator function of a set, A, is denoted by l and is defined as follows...
The indicator function of a set, A, is denoted by l and is defined as follows (a) Show that if A and B are any two sets, then the indicator function, ln, of (b) IfA and B are disjoint (i.e An B is empty), show that luB is the sum, 1, if x E A A n B, is the product lalB.i.e Show that lanB (x) =ム(x)h(x) for every x. ム+1B. İ.e Show that 1AUB (x)-4(x) + 1B (x) for...
Let Xi,... ,Xn be i.i.d with pdf θνθ θ+1 where I(.) denotes the indicator function. (a) Find a 2-dimensional sufficient statistic for the mode (b) Suppose θ is a known constant. Find the MLE for v....
Let Xi,... ,Xn be i.i.d with pdf θνθ θ+1 where I(.) denotes the indicator function. (a) Find a 2-dimensional sufficient statistic for the mode (b) Suppose θ is a known constant. Find the MLE for v. (d) Suppose v-1. Find the MLE for and determine its asymptotic distribution. Carefully justify your answer and state any theorems that you use. (e) Suppose1. Find the asymptotic distribution of the MLE estimator of exp[-
Let Xi,... ,Xn be i.i.d with pdf θνθ θ+1...
*5. A function f defined on an interval I = {x: a <x<b> is called increasing...
*5. A function f defined on an interval I = {x: a <x<b> is called increasing f(x) > f(x2) whenever xi > X2 where x1, x2 €1. Suppose that has the inter- mediate-value property: that is, for each number c between f(a) and f(b) there is an x, el such that f(x) = c. Show that a functionſ which is increasing and has the intermediate-value property must be continuous. This is from my real analysis textbook, We are establishing the...
Definition 1. A function f(x) defined on (-L, L] is called piece-wise continuous if there are finitely many points xo =-L < x1 < x2 < < xn-L such that f is continuous on (xi, i+1) a...
Definition 1. A function f(x) defined on (-L, L] is called piece-wise continuous if there are finitely many points xo =-L < x1 < x2 < < xn-L such that f is continuous on (xi, i+1) and so that the limits lim f(z) and lim f(x) both crist for each a,. To save space we write lin. f(x) = f(zi-) ェ→z, lim, f(x) = f(zit), ェ→ Sub-problem 5. Let f(x)-x on (-2,-1), f(x) = 1 on (-1,0) and f(x)--z on...
For the system shown in Fig. 1, solve the following problems. (a) Find the transfer function, G(s...
For the system shown in Fig. 1, solve the following problems. (a) Find the transfer function, G(s)X2 (s)/F(s) (b) Does the system oscillate with a unit step input (f (t))? Explain the reason (c) Decide if the system(x2 (t)) is stable with a unit step input (f (t))? Explain the reason 1. 320) 8 kg 2 N/m 4N-s/m 2N-s/m Fig. 1 2. There are two suspensions for a car as shown in Fig. 2 (a) Find the equations of each...
The indicator function of a set, A, is denoted by l and is defined as follows (a) Show that if A and B are any two sets, then the indicator function, ln, of (b) IfA and B are disjoint (i.e An B is empty), show that luB is the sum, 1, if x E A A n B, is the product lalB.i.e Show that lanB (x) =ム(x)h(x) for every x. ム+1B. İ.e Show that 1AUB (x)-4(x) + 1B (x) for...
Let Xi,... ,Xn be i.i.d with pdf θνθ θ+1 where I(.) denotes the indicator function. (a) Find a 2-dimensional sufficient statistic for the mode (b) Suppose θ is a known constant. Find the MLE for v. (d) Suppose v-1. Find the MLE for and determine its asymptotic distribution. Carefully justify your answer and state any theorems that you use. (e) Suppose1. Find the asymptotic distribution of the MLE estimator of exp[-
Let Xi,... ,Xn be i.i.d with pdf θνθ θ+1...
*5. A function f defined on an interval I = {x: a <x<b> is called increasing f(x) > f(x2) whenever xi > X2 where x1, x2 €1. Suppose that has the inter- mediate-value property: that is, for each number c between f(a) and f(b) there is an x, el such that f(x) = c. Show that a functionſ which is increasing and has the intermediate-value property must be continuous. This is from my real analysis textbook, We are establishing the...
Definition 1. A function f(x) defined on (-L, L] is called piece-wise continuous if there are finitely many points xo =-L < x1 < x2 < < xn-L such that f is continuous on (xi, i+1) and so that the limits lim f(z) and lim f(x) both crist for each a,. To save space we write lin. f(x) = f(zi-) ェ→z, lim, f(x) = f(zit), ェ→ Sub-problem 5. Let f(x)-x on (-2,-1), f(x) = 1 on (-1,0) and f(x)--z on...
For the system shown in Fig. 1, solve the following problems. (a) Find the transfer function, G(s)X2 (s)/F(s) (b) Does the system oscillate with a unit step input (f (t))? Explain the reason (c) Decide if the system(x2 (t)) is stable with a unit step input (f (t))? Explain the reason 1. 320) 8 kg 2 N/m 4N-s/m 2N-s/m Fig. 1 2. There are two suspensions for a car as shown in Fig. 2 (a) Find the equations of each...
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