A lamina occupies the part of the rectangle 0≤x≤40≤x≤4,
0≤y≤30≤y≤3 and the density at each point...
A lamina occupies the part of the rectangle 0≤x≤40≤x≤4,
0≤y≤30≤y≤3 and the density at each point is given by the function
ρ(x,y)=3x+7y+4ρ(x,y)=3x+7y+4.
A lamina with constant density
ρ(x, y) =
ρ
occupies the given region. Find the moments of inertia
Ix
and
Iy
and the radii of gyration and .
The part of the disk
x2 +
y2 ≤
a2
in the first quadrant
A lamina occupies the region inside the circle x^2 + y^2 = 6x,
but outside the circle x^2 + y^2 =9.
Find the center of mass if the density at any point is inversely
proportional to its distance from the origin.
A lamina occupies the region inside the circle x^2 + y^2 = 6x,
but outside the circle x^2 + y^2 =9.
Find the center of mass if the density at any point is inversely
proportional to its distance from the origin.
f(x,y)=30(1-y)^2*x*e^(-x/y). x>0. 0<y<1.
a). show that f(y) the marginal density function of Y is a Beta
random variable with parameters alfa=3 and Beta=3.
b). show that f(x|y) the conditional density function of X given
Y=y is a Gamma random variable with parameters alfa=2 and
beta=y.
c). set up how would you find P(1<X<3|Y=.5). you do not
have to do any calculations
Given the joint density function of X and Y as
fX,Y(x,y) = cx2 + xy/3
0 <x <1 and 0 < y < 2.
complete work shading appropriate regions for all integral
calculations.
Find the expected value of Z =
e(s1X+s2Y) where s1 and
s2 are any constants.
4. The joint density function of (X, Y ) is
f(x,y)=2(x+y), 0≤y≤x≤1
. Find the correlation coefficient ρX,Y
.
5. The height of female students in KU follows a normal
distribution with mean 165.3 cm and s.d. 7.3cm. The height of male
students in KU follows a normal distribution with mean 175.2 cm and
s.d. 9.2cm. What is the probability that a random female student is
taller than a male student in KU?
Solve Laplace's equation inside a rectangle 0 ≤ x ≤ L, 0 ≤ y ≤
H, with the following boundary conditions [Hint: Separate
variables. If there are two homogeneous boundary conditions in y,
let u(x,y) = h(x)∅(y), and if there are two homogeneous boundary
conditions in x, let u(x,y) = ∅(x)h(y).]:
∂u/∂x(0,y) = 0
∂u/∂x(L,y) = 0
u(x,0) = 0
u(x,H) = f(x)
Solve Laplace’s equation wxx + wyy = 0 on
the rectangle R = {(x, y) : 0 ≤ x ≤ a, 0 ≤ y ≤ b} subject to the
boundary conditions w(x, 0) = 0, w(x, b) = 0, w(0, y) =
f1(y), w(a, y) = f2(y). Include
coefficient formulas.
a. For the following probability density
function:
f(X)=
3/4 (2X-X^2 ) 0 ≤ X ≤ 2
=
0 otherwise
find
its expectation and variance.
b. The two regression lines are 2X - 3Y + 6 = 0 and 4Y – 5X- 8
=0 , compute mean of X and mean of Y. Find correlation coefficient
r , estimate y for x =3 and x for y = 3.