![Prove that if X 20, Y 2 0 and 0 p1, then E(X +Y)] Show that for any real numbers x > 0 and y > 0, E(X)E(YP). HINT: Here is how you can show the above formula holds. Start off by letting 0y. Use the fact that the function g(z) - z is concave-down (i.e., spills water) on (0, oo) and is thus bounded above by its tangent line at any particular point. Find the tanget line at the point z = y. This tangent line will be an upper bound for g(z), so write out the resulting inequality. Then let z+y.](http://img.homeworklib.com/questions/27039c90-b45c-11ec-a458-1705dc16bb47.png?x-oss-process=image/resize,w_560)
Homework Answers
Given 
. So 
First find the tangent line to the curve 
. Take the derivative,
The equation of tangent at 
is
Since
, is concave down, the tangent is upper bound for
. Thus we have
Let 
, then since
and
we have 
and set 
The above inequality becomes,
For 
,
we can write,
Take expectations on both sides,
The proof is complete.
2. Let f(x,y) = e-r-u, 0 < x < oo, 0 < y < oo, zero...
2. Let f(x,y) = e-r-u, 0 < x < oo, 0 < y < oo, zero elsewhere, be the pdf of X and Y. Then if Z = X + Y, compute (a) P(Z 0). (b) P(Z 6) (c) P(Z 2) (d) What is the pdf of Z?
part (c) 7.23. Let y(x) = n²x e-nx. (a) Show that lim, - fn(x)=0 for all...
part (c)
7.23. Let y(x) = n²x e-nx. (a) Show that lim, - fn(x)=0 for all x > 0. (Hint: Treat x = 0 as for x > 0 you can use L'Hospital's rule (Theorem A.11) - but remember that n is the variable, not x.) (b) Find lim - So fn(x)dx. (Hint: The answer is not 0.) (c) Why doesn't your answer to part (b) violate Proposition 7.27 Proposition 7.27. Suppose f. : G C is continuous, for n...
2. Suppose that is an exponential random variable with pdf f(y)= e), y>0. a. Use Chebyshev's Inequality to get an up...
2. Suppose that is an exponential random variable with pdf f(y)= e), y>0. a. Use Chebyshev's Inequality to get an upper bound for the probability that takes on a value more than two standard deviations away from the mean. b. Use the given pdf to compute the exact probability that takes on a value more than two standard deviations away from the mean.
0 intersect only at (0,0) g(r)at z arctan(3z) Show that the graph y f(x) and its tangent line y p...
0 intersect only at (0,0) g(r)at z arctan(3z) Show that the graph y f(x) and its tangent line y po Consider the ftunction f(x) Intermediate steps: 1) The lIne tangent to y f(x)atz -0isy g(x) where g(r) 9(a)- 2Let H(x) f(x) - 9(x) The derivative ot H (x)s H'(z) = which is zero only when x = Rolle's theorem to H (x) on the interval [ri, 0]. Get a contradiction. 4) Now assume that we have zp O where f(2)-9(T2)...
1 point) Integratef(x, y, z) 6xz over the region in the first octant (x,y, z 0) above the parabol...
Please try helping with all three questions.......please
1 point) Integratef(x, y, z) 6xz over the region in the first octant (x,y, z 0) above the parabolic cylinder z = y2 and below the paraboloid Answer Find the volume of the solid in R3 bounded by y-x2 , x-уг, z-x + y + 24, and Z-0. Consider the triple integral fsPw xyz2 dV, where W is the region bounded by Write the triple integral as an iterated integral in the order...
2. We say that two curves intersect orthogonally if they intersect and their tangent lines are orthogonal at each point in the intersection. For example, the curve y = 0 intersects the curve x2 + y2-...
2. We say that two curves intersect orthogonally if they intersect and their tangent lines are orthogonal at each point in the intersection. For example, the curve y = 0 intersects the curve x2 + y2-1 orthogonally at (-1,0) and (1,0). Let H be the set of curves y2b with b ER. (a) Prove that the tangent line of each curve in H at a point (r, y) with y / 0 has slope (b) Let y -f(x) be a...
20. The volume of the region bounded by the graphs of x = 0, y =...
20. The volume of the region bounded by the graphs of x = 0, y = 0, z = 0 and x + y + z = 2 is: a) b) c) d)4 e) none of these
For the Hamiltonian syste m we did in class: 2. 3 Ic (1) Show that it's a Hamiltonian system with a Hamiltonian function (2) Show that for each c > 0, {(z,y) є R2 . H(z,y) c} is a bounded inva...
For the Hamiltonian syste m we did in class: 2. 3 Ic (1) Show that it's a Hamiltonian system with a Hamiltonian function (2) Show that for each c > 0, {(z,y) є R2 . H(z,y) c} is a bounded invariant set of the dynamical system (in fact, it's also closed) (3) Find all the equilibria of this system. Show that H-() is made up of one equilibium point and two homoclinic orbits attached to it. (4) Sketch the invariant...
Solve the following differential equation on the interval (0, oo) with conditions lim y(e) im y(x)-0 x→0+ y(z) = 0 an and Write your answer by Bessel functions Solve the following differentia...
Solve the following differential equation on the interval (0, oo) with conditions lim y(e) im y(x)-0 x→0+ y(z) = 0 an and Write your answer by Bessel functions
Solve the following differential equation on the interval (0, oo) with conditions lim y(e) im y(x)-0 x→0+ y(z) = 0 an and Write your answer by Bessel functions
Question 20 The solution to the linear inequality − 10 x + 2 y > −...
Question 20 The solution to the linear inequality − 10 x + 2 y > − 4 will be the half-plane lying a) on and below the solid line − 10 x + 2 y = − 4. b) below the solid line − 10 x − 2 y = − 4. c) above the dashed line − 10 x + 2 y = − 4. d) on and above the solid line − 10 x + 2 y = − 4. e) below...
2. Let f(x,y) = e-r-u, 0 < x < oo, 0 < y < oo, zero elsewhere, be the pdf of X and Y. Then if Z = X + Y, compute (a) P(Z 0). (b) P(Z 6) (c) P(Z 2) (d) What is the pdf of Z?
part (c)
7.23. Let y(x) = n²x e-nx. (a) Show that lim, - fn(x)=0 for all x > 0. (Hint: Treat x = 0 as for x > 0 you can use L'Hospital's rule (Theorem A.11) - but remember that n is the variable, not x.) (b) Find lim - So fn(x)dx. (Hint: The answer is not 0.) (c) Why doesn't your answer to part (b) violate Proposition 7.27 Proposition 7.27. Suppose f. : G C is continuous, for n...
2. Suppose that is an exponential random variable with pdf f(y)= e), y>0. a. Use Chebyshev's Inequality to get an upper bound for the probability that takes on a value more than two standard deviations away from the mean. b. Use the given pdf to compute the exact probability that takes on a value more than two standard deviations away from the mean.
0 intersect only at (0,0) g(r)at z arctan(3z) Show that the graph y f(x) and its tangent line y po Consider the ftunction f(x) Intermediate steps: 1) The lIne tangent to y f(x)atz -0isy g(x) where g(r) 9(a)- 2Let H(x) f(x) - 9(x) The derivative ot H (x)s H'(z) = which is zero only when x = Rolle's theorem to H (x) on the interval [ri, 0]. Get a contradiction. 4) Now assume that we have zp O where f(2)-9(T2)...
Please try helping with all three questions.......please
1 point) Integratef(x, y, z) 6xz over the region in the first octant (x,y, z 0) above the parabolic cylinder z = y2 and below the paraboloid Answer Find the volume of the solid in R3 bounded by y-x2 , x-уг, z-x + y + 24, and Z-0. Consider the triple integral fsPw xyz2 dV, where W is the region bounded by Write the triple integral as an iterated integral in the order...
2. We say that two curves intersect orthogonally if they intersect and their tangent lines are orthogonal at each point in the intersection. For example, the curve y = 0 intersects the curve x2 + y2-1 orthogonally at (-1,0) and (1,0). Let H be the set of curves y2b with b ER. (a) Prove that the tangent line of each curve in H at a point (r, y) with y / 0 has slope (b) Let y -f(x) be a...
20. The volume of the region bounded by the graphs of x = 0, y = 0, z = 0 and x + y + z = 2 is: a) b) c) d)4 e) none of these
For the Hamiltonian syste m we did in class: 2. 3 Ic (1) Show that it's a Hamiltonian system with a Hamiltonian function (2) Show that for each c > 0, {(z,y) є R2 . H(z,y) c} is a bounded invariant set of the dynamical system (in fact, it's also closed) (3) Find all the equilibria of this system. Show that H-() is made up of one equilibium point and two homoclinic orbits attached to it. (4) Sketch the invariant...
Solve the following differential equation on the interval (0, oo) with conditions lim y(e) im y(x)-0 x→0+ y(z) = 0 an and Write your answer by Bessel functions
Solve the following differential equation on the interval (0, oo) with conditions lim y(e) im y(x)-0 x→0+ y(z) = 0 an and Write your answer by Bessel functions
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