5. Suppose XExp(A). (a) [5 pts] Show that E(X) 1/A. [Hint: You can directly use the...
5. Suppose XExp(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 +s | X > s) = P(X > t) for s, t > 0, [Hint: You can directly use the tail probablity P(X >) e for 0.]
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.
hint
This exercise 5 to use the definition of Riemann integral
F. Let f : [a, b] → R be a bounded function. Suppose there exist a sequence of partitions {Pk} of [a, b] such that lim (U(Pk, f) – L (Pk,f)) = 0. k20 Show that f is Riemann integrable and that Så f = lim (U(P«, f)) = lim (L (Pk,f)). k- k0 1,0 < x <1 - Suppose f : [-1, 1] → R is defined as...
5. Let X ~ Exp(A) with λ unknown, and suppose X1,X2 is a random sample of size 2, Show that M-X (Hint: During your journey, you' need the help of the gamma distribution, the gamma function, and the knowledge that Г(1/2-ут) X1 X2 is a biased estimator of - and modify it to create an unbiased estimator
Show that the sum of the observations of a random sample of size n from gamma distribution with parameters 1 and θ (so f(x:0)-e-",x > 0 ) is sufficient for θ, using the definition ofsuficiency. Then show that the mle of θ is a function of the sufficient x10 statistic.
Show that the sum of the observations of a random sample of size n from gamma distribution with parameters 1 and θ (so f(x:0)-e-",x > 0 ) is sufficient for...
Show that the sum of the observations of a random sample of size n from gamma distribution with parameters 1 and θ (so f(x:0)-e-re, x > 0 ) is sufficient for θ, using x/θ the definition ofsuficiency. Then show that the mle of θ is a function of the sufficient statistic.
Show that the sum of the observations of a random sample of size n from gamma distribution with parameters 1 and θ (so f(x:0)-e-re, x > 0 ) is...
Problem 2 Show that e) Hint: If you cannot get the desired estimate directly, try using domain decomposition.]
Question 1
1. [5 pts] Give a complete definition of lim f(x) = -oo if... 2. [25 pts] Give an example of each of the following, or state one or more theorems which show that such an example is impossible: a. A countable collection of nonempty closed proper subsets of R whose union is open. b. A nonempty bounded subset of R with no cluster points. c. A convergent sequence with two convergent subsequences with distinct limits. d. A function...
1. Find a matrix A so that A | y for all z, y, z E R. What are the dimensions of A? 2y +2z (The dimensions of an m x n matrix are "m × n.) for all R2. Find a matrix A so that T-LA (that is. Τ(x) = Ax for all fe R2). and all vectorsR2. Do not assume any properties of the dot product, beyond the definition. (Hint write Aa21 a22and x 2. Let T: IR2R2...
we use this definition
5. [3 points Prove that the function f(x) = - , is continuous at := -1. You should give a proof that is directly based on the definition of continuity. Solution: You can type your solutions here. teso Isso sit & lx-xokę => 1 F(X) - F(Xoll LE