*(4) (a) Prove that if p=(x,y) is in the set where y<x and if
r=distance from p to the line y=x then the ball about p of radius r
does not intersect with the line y=x.
(b) Prove that the set where y<c is an open set.
Justify your answer
5. Equations of the form y’ = P(x)*y^2 + Q(x)*y + R(x) are
called Riccati equations.
i) If we know a solution y = φ(x) of this equation, then any
other solution
can be written in the form y(x) = φ(x)+ 1/v(x), where v(x) is an
unknown
function which satisfies a certain linear equation. Using the
fact that
φ and y both solve the above Riccati equation, find the
differential
equation that v satisfies.
ii) Consider the equation 3y’ +...
5. Equations of the form y’ = P(x)*y^2 + Q(x)*y + R(x) are
called Riccati equations.
i) If we know a solution y = φ(x) of this equation, then any
other solution
can be written in the form y(x) = φ(x)+ 1/v(x), where v(x) is an
unknown
function which satisfies a certain linear equation. Using the
fact that
φ and y both solve the above Riccati equation, find the
differential
equation that v satisfies.
ii) Consider the equation 3y’ +...
f:
R[x] to R is the map defined as f(p(x))=p(2) for any polynomial
p(x) in R[x]. show that f is
1) a homomorphism
2) Ker(f)=(x-2)R[x]
3) prove that R[x]/Ker(f) is an isomorphism with R.
(R in this case is the Reals so
R[x]=a0+a1x+a1x^2...anx^n)
Please Answer the following with as much detail as possible.
Thanks!
What does a p-value indicate?
How does the standard deviation differ from the standard error?
(Hint: Don't just describe the difference in the formulas, explain
why the formulas are different)
How do you know if the results of a sample can be generalized
to the entire population?
Please Consider the function f : R -> R given by f(x, y) = (2
- y, 2 - x).
(a) Prove that f is an isometry.
(b) Draw the triangle with vertices A = (1, 2), B = (3, 1), C =
(3, 2), and the triangle with vertices f(A), f(B), f(C).
(c) Is f a rotation, a translation, or a glide reflection?
Explain your answer.
Suppose that X ~ NB(r, p) [negative binomial distribution] and
that Y ~ B(n, p) [binomial].
a. Give a probabilistic argument to show that P(X > n) = P(Y
< r).
b. Use the FPF to express the equality in part (a) in terms of
PMFs.
c. Using the complementation rule, how many terms of the PMF of
X must be evaluated to determine P(X > n)?
d. How many terms of the PMF of Y must be evaluated to...
let p ( x , y )be a joint pmf of Xand Y.
p ( x , y )
y = 0
y = 1
y = 2
x = 0
.12
.10
.08
x = 1
.13
.17
.10
x = 2
.15
.15
0
(a) Find the marginal pmf's of Xand Y.
(b) Determine whether Xand Yare independent.
c) Find Correlation (X,Y)