Question

Let fx,y (x,y) = 3 e^-(x+y) for 0 < x <1/2y and y>0. a) Find f x(x) and f y( y) . b) Write out the integral necessary to find , Fx,y ( u v) . DO NOT EVALUATE THE INTEGRAL.

Answer #1

; 0 < x < 1/2; y > 0..

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Let f (x, y) = 100 sin(π(x−2y))/(1+x^2+y^2) . Find the
directional derivative of f 1+x^2+y^2 at the point (10, 6) in the
direction of: (a) u = 3 i − 2 j (b) v = −i + 4 j

consider the joint density function
Fx,y,za (x,y,z)=(x+y)e^(-z) where 0<x<1, 0<y<1,
z>0
find the marginal density of z : fz (z).
hint. figure out which common distribution Z follows and report
the rate parameter
integral (x+y)e^(-z) dz
(x+y)(-e^(-z) + C
is my answer 1. ???

Let X and Y be continuous random variables with joint density
function f(x,y) and marginal density functions fX(x) and fY(y)
respectively. Further, the support for both of these marginal
density functions is the interval (0,1).
Which of the following statements is always true? (Note there
may be more than one)
E[X^2Y^3]=(∫0 TO 1 x^2 dx)(∫0 TO 1 y^3dy)
E[X^2Y^3]=∫0 TO 1∫0 TO 1x^2y^3 f(x,y) dy dx
E[Y^3]=∫0 TO 1 y^3 fX(x) dx
E[XY]=(∫0 TO 1 x fX(x)...

4. Let X and Y be random variables having joint probability
density function (pdf) f(x, y) = 4/7 (xy − y), 4 < x < 5 and
0 < y < 1
(a) Find the marginal density fY (y).
(b) Show that the marginal density, fY (y), integrates to 1
(i.e., it is a density.)
(c) Find fX|Y (x|y), the conditional density of X given Y =
y.
(d) Show that fX|Y (x|y) is actually a pdf (i.e., it integrates...

1. for 0<= x <=3 0<=x<=1 f(x,y) = k(x^2y+ xy^2)
a. Find K joint probablity density function.
b. Find marginal distribution respect to x
c. Find the marginal distribution respect to y
d. compute E(x) and E(y) e. compute E(xy)
f. Find the covariance and interpret the result.

Let joint CDF Fx,y (x,y) = сxy(x2 + y2)
for 0 ≤ x ≤ 1, 0 ≤ y ≤ 1.
Find а) constant с.
b) Fx|y (x|y) for x = 0.5, y = 0.5.

Let s = f(x; y; z) and x = x(u; v; w); y = y(u; v; w); z = z(u;
v; w). To calculate ∂s ∂u (u = 1, v = 2, w = 3), which of the
following pieces of information do you not need?
I. f(1, 2, 3) = 5
II. f(7, 8, 9) = 6
III. x(1, 2, 3) = 7
IV. y(1, 2, 3) = 8
V. z(1, 2, 3) = 9
VI. fx(1, 2, 3)...

Let X and Y have joint density f given by f(x, y) = cxy 0 ≤ y ≤
x, 0 ≤ x ≤ 1.
(a) Determine the normalization constant c.
(b) Determine P(X + 2Y ≤ 1).
(c) Find E(X|Y = y).
(d) Find E(X).

Ex-12.3
Let joint CDF FX,Y (x,y) = сxy(x^2 + y^2) for
0<=x<=1, 0<=y<=1.
Find а) constant с.
b) Fx|y (X,Y) for x = 0.5, y = 0.5.

The real part of a f (z) complex function is given as
(x,y)=y^3-3x^2y. Show the harmonic function u(x,y) and find the
expressions v(x,y) and f(z). Calculate f'(1+2i) and write x+iy
algebraically.

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