If E [exp(aX)] exists for a given constant a, then
show that for t > 0
(a) exp(−at)P (X > t) < E [exp(aX )] , if a > 0.
(b) exp(−at)P (X < t) < E [exp(aX )] , if a < 0.
Note-if
there is any understanding problem regarding this please feel free
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If E [exp(aX)] exists for a given constant a, then show that for t > 0...
Extra Credit Question:[4+4=8 pts) If E [exp(aX)] exists for a given constant a, then show that for t > 0 (a) exp(-at)P(X >t) <E [exp(aX)], if a > 0. (b) exp(-at)P(X<t) <E [exp(aX)], if a <0.
Extra Credit Question:[4+4=8 pts) If E [exp(aX)] exists for a given constant a, then show that for to (a) exp(-at)P(x >t) <E (exp(aX)], if a > 0. (b) exp(-at)P(X <t) <E (exp(aX)], if a < 0.
Let X1,...X be i.i.d with density f()(1/0)exp(-/0) for r >0 and 0> 0. a. Find the pitman estimator of 0 b. Show that the pitman estimator has smaller risk than the UMVUE of when the loss function is (t-0)2 02 Suppose C. f(x)= 0exp(-0x) and that 0 has a gamma prior with parameters a and p, find the Bayes estimator of 0 d. Find the minimum Bayes risk e. Find the minimax estimator of 0 if one exists. 1
Let...
Problem 4.26
194 Chap. 4 Duality theory Exercise 4.26 Let A be a given matrix. Show that exactly one of the following alternatives must hold. (a) There exists some x 0 such that Ax = 0, x > 0. (b) There exists some p such that p'A>0'. Exercise 4.27 Let A be a given matrix. Show that the following two state- ments are equivalent. (a) Every vector such that Ax > 0 and x > 0 rnust satisfy x1 =...
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3. Let f: RP-R (a) If f(x)-Ax + b,...
10. The moment generating function of the random variable X is given by My(t) = exp{2e* – 2} and that of Y by My(t) = fet +. Assuming that X and Y are independent, find (a) P{X + Y = 2). (b) P{XY = 0}. (c) E(XY).
5. Suppose X ~Exp(A). (a) [5 pts] Show that E(X) 1/A. [Hint: You can directly use the definition and properties of a gamma function.] (b) [5 pts] Prove that P(x >t+ |xs) P(x > t) for s,t>0. [Hint: You can directly use the tail probability P(X > x) = e-k for x > 0.
(3) Suppose that f E C'((0, 1]). Given e > 0, prove that there exists a polynomial p such that lf-plloo -p'| <E
(3) Suppose that f E C'((0, 1]). Given e > 0, prove that there exists a polynomial p such that lf-plloo -p'|
2 (1) For z E C, define exp z - n-0 (a) Prove that the infinite series converges absolutely for z E C (b) Prove that when z є R, the definition of exp z given above is consistent with the one given in problem (2a), assignment 16. Definition from Problem (2a): L(x(1/t)dt E(z) = L-1 (z)
2 (1) For z E C, define exp z - n-0 (a) Prove that the infinite series converges absolutely for z E C...