help me with these two probability questions. 12. (3 points) Let f(x) = kx3 +2 -1 for 0 <3 <1, and f(2)= 0 for [0, 1], where k is some unknown constant. What value of k will make f a valid probability density function? A. 1/4 B. 1/3 C. 1 D. 2 -xel + Ž - 1=1 | LEH , E = Ž- at zlik Ks6 나 E. 4 F. 6 G. No such value of k exists + C....
(1 point) Let f(x) = 0 if x < -4 5 if – 4 < x < 0 -3 if 0 < x < 3 0 if x 2 3 and g(x) = Los f(t)dt Determine the value of each of the following: (a) g(-8) = 0 (b) g(-3) = 5 (c) g(1) = (d) g(4) = (e) The absolute maximum of g(x) occurs when x = 0 and is the value It may be helpful to make a graph...
Let f(x) = 2x4 +x4cos(1/x) for x ̸= 0 and f(0) = 0. Show that 0 is a global minimum x for f but for every neighbourhood V of 0 there exists x,y ∈ V such that f′(x) > 0 and f′(y) < 0.
Let f(x) = cxe-x if x 20 and f(x) = 0 if x < 0. (a) For what value of c is fa probability density function? (b) For that value of c, find P(1<x< 4). 0.368
Exercice 1 We consider the function f(x) = 2 #0 and for r > 0. let S, = {€ C/2 = r} with positive orientation. For 0 < <R, we denote by r the curve consisting of SRUT-R,-€) US, UL, R), where S = {z E C/121 = } with negative orientation. 1. Prove that o = [513)dz = [5(=)dz + [s()de – [ (dz + 1" $(x)dr.
LI CONTINUOUS DIST Let X be a random variable with pdf -cx, -2<x<0 f(x)={cx, 0<x<2 otherwise where c is a constant. a. Find the value of c. b. Find the mean of X. C. Find the variance of X. d. Find P(-1 < X < 2). e. Find P(X>1/2). f. Find the third quartile.
4. Let f(x, y) = 6x, x > 0, y > 0, x +y < 1. Find P(X< }). (a) .3827 (b) .2593 (c).2126 (d).1875 (e).1383
·J (I) < 0 for all such y. (Hint: let g(x)--f(x) and use part (a)) 3. In this problem, we prove the Intermedinte Value Theorem. Let Intermediate Value Theorem. Let f : [a → R be continuous, and suppose f(a) < 0 and f(b) >0. Define S = {t E [a, b] : f(z) < 0 for allェE [a,t)) (a) Prove that s is nonempty and bounded above. Deduce that c= sup S exists, and that astst (b) Use Problem...
7.2 Let X have density f(x) = cx for 0 < x < 2 and f(x) = 0 for other values of x. a. What is c? b. What is F(x)? c. What are E[X] and Var[x]? 7.3 Let X have density f(x) = cx(1 - x) for 0 sxs 1 and f(x) = 0 for other values of x. a. What is c? b. What is F(x)? c. What are E[X] and Var[x]?
Suppose f'(x) = -1/3(x+3). On what open interval(s) is f(x) decreasing 0-3 < < 0 0-3 <x<0 0 - < I< -3 0 - < ?<-3 and 0 < x < 0