8. Let f (x) e, 0 > 0; x> 0 (1 1 +e (a) Show that f(x) is a probability density function (b) Find P(X> x) (c) Find the failure rate function of X
Suppose that f is integrable on (a, b) and define (f(x) if f(x) > 0 f+(x) = 3 and f (2)= if f(x) < 0, Show that f+ and f- are integrable on (a, b), and If(x) if f(x) > 0, if f(x) < 0. cb Sisleyde = [* p*(e) ds + [°r(a)di. | f(x) dx = | f+(x) dx + 1 f (x) dx.
Prove that if ? is integrable on [?, ?] and ?(?) ≥ 0 for all ?
in [?, ?], then
[ f(x)dx > 0 7. Prove that if f is integrable on [a, b] and f(x) > 0 for all x in [a, b], then sof(x)dx > 0.
7. Let X1,... , Xn be iid based on f(x; 6) -22e-z?/e where x > 0. Show that θ=-yx? is efficient
Problem 2. Let f be a self-map on a set X. For x,y e X define x ~ y if and only if f"(x) = f(y) for some integers n, m > 0. Show that ~ is an equivalence relation.
Let Xi X, lid f(r 0) with f(r:0)-e ( e) for r > ? and ? e R. (a) Find the MLE of ? (c) Using the prior density ?(0)-e-91(0,0)( ?), find the Bayesian estimator of ?
check if
e-1/4/ f(x) if x > 0 if x < 0 is differentiable at 0.
Let F be a field of characteristic p > 0. Show that f = t4 +1 € F[t] is not irreducible. Let K be a splitting field of f over F. Determine which finite field F must contain so that K = F.
Exercise 1. Let f : R R be differentiable on la, b, where a, b R and a < b, and let f be continuous on [a, b]. Show that for every e> 0 there exists a 6 > 0 such that the inequality f(x)- f(c) T-C holds for all c, x E [a, 히 satisfying 0 < |c-x| < δ
Vx+1-1 Evaluate: lim x>0 х Please solve it in detail and show all your steps./