The random vector ( X , Y ) is said to be uniformly distributed over a region R in the plane if, for some constant c, its joint density is f ( x , y ) = { c if ( x , y ) ∈ R 0 otherwise a. Show that 1 c = area of regopm R . Suppose that ( X , Y ) is uniformly distributed over the square centered at (0, 0) and with sides of length 2. b. Show that X and Y are independent, with each being distributed uniformly over ( − 1 , 1 ) . c. What is the probability that ( X , Y ) lies in the circle of radius 1 centered at the origin? That is, find P { X 2 + Y 2 ≤ 1 } .
The random vector ( X , Y ) is said to be uniformly distributed over a region R in the plane if, for some constant c, its joint density is f ( x , y ) = { c if ( x , y ) ∈ R 0 otherwise a. Show that 1 c = area of regopm R . Suppose that ( X , Y ) is uniformly distributed over the square centered at (0, 0) and with sides of length 2. b. Show that X and Y are independent, with each being distributed uniformly over ( − 1 , 1 ) . c. What is the probability that ( X , Y ) lies in the circle of radius 1 centered at the origin? That is, find P { X 2 + Y 2 ≤ 1 } .
The random vector
(
X
,
Y
)
is said to be uniformly distributed over a region R in the plane if, for some constant c, its joint density is
f
(
x
,
y
)
=
{
c
if
(
x
,
y
)
∈
R
0
otherwise
a. Show that
1
c
=
area of regopm R
. Suppose that
(
X
,
Y
)
is uniformly distributed over the square centered at (0, 0) and with sides of length 2.
b. Show that X and Y are independent, with each being distributed uniformly over
(
−
1
,
1
)
.
c. What is the probability that
(
X
,
Y
)
lies in the circle of radius 1 centered at the origin? That is, find
P
{
X
2
+
Y
2
≤
1
}
.
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
Find the cdf and density of X2 where X ∼ N (μ, σ2).
A lamina occupies a triangular region D in the xy-plane. D is enclosed by the lines
|x = 0, y = x, and 2x + y = 6. If density p(x, y) = x+y, what is the mass and center of mass?
3. A region V is defined by the quartersphere x² + y² + zx² = 16, z ≥ 0, y ≥ 0
and the planes z = 0, y = 0. A vector field
F = zyi + y²j+k
exists throughout and on the boundary of the region. Verify the GAUSS DIVER-
GENCE THEOREM for the region.
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