if X1, X2 have the joint pdf
f(x1, x2) = 4x1(1-x2) ,
0<x1<1
0<x2<1
and...
if X1, X2 have the joint pdf
f(x1, x2) = 4x1(1-x2) ,
0<x1<1
0<x2<1
and 0,
otherwise
1- Find the probability
P(0<X1<1/3 , 0<X2<1/3)
2- For the same joint pdf, calculate
E(X1X2) and E(X1 +
X2)
3- Calculate
Var(X1X2)
Consider the Rosenbrock function, f(x) = 100(x2 -
x12)2 + (1
-x1)2. Let x*=(1,1), the minimum...
Consider the Rosenbrock function, f(x) = 100(x2 -
x12)2 + (1
-x1)2. Let x*=(1,1), the minimum of the
function.
Let r(x) be the second order Taylor series for f(x) about the
base point x*, r(x) will be a quadratic, and therefore can be
written:
r(x) = A11(x1-x1*)2
+ 2A12(x1-x1*)(x2-x2*) +
A22(x2-x2*)2 +
b1(x1-x1*) +
b2(x2-x2*) + c
Find all the coefficients in the formula for r(x) - What is
A11, A12, A22, b1,
b2, and c?
Let X =( X1,
X2, X3 ) have the joint pdf
f(x1, x2,
x3)=60x1x22, where
x1...
Let X =( X1,
X2, X3 ) have the joint pdf
f(x1, x2,
x3)=60x1x22, where
x1 + x2 + x3=1 and
xi >0 for i = 1,2,3. find the
distribution of X1 ? Find
E(X1).
Let f(x1, x2) = 1 , 0 ≤ x1 ≤ 1 , 0 ≤ x2 ≤...
Let f(x1, x2) = 1 , 0 ≤ x1 ≤ 1 , 0 ≤ x2 ≤ 1 be the joint pdf of
X1 and X2 . Y1 = X1 + X2 and Y2 = X2 .
(a) E(Y1) .
(b) Var(Y1)
(c) Consider the marginal pdf of Y1 , g(y1) . What is value of
g(y1) where y1 = 1/3 and y1 = 6/4 ?
(i) Find the probability P(0<X1<1/3 ,
0<X2<1/3) where X1, X2 have
the joint pdf
f(x1,
x2)...
(i) Find the probability P(0<X1<1/3 ,
0<X2<1/3) where X1, X2 have
the joint pdf
f(x1,
x2) = 4x1(1-x2)
, 0<x1<1 0<x2<1
0, otherwise
(ii) For the same joint pdf, calculate
E(X1X2) and E(X1+
X2)
(iii) Calculate Var(X1X2)
Let T ∈ L(R2) be the linear transformation T(x1, x2) = (3x1 +
2x2, −4x1 −...
Let T ∈ L(R2) be the linear transformation T(x1, x2) = (3x1 +
2x2, −4x1 − 3x2), v = (1, −1), and p(z) = z^2 − 3z + 2. Compute
p(T), show that p(T)v = 0, and show that NOT all the roots of p(z)
are eigenvalues of T.
Let (X1,d1) and (X2,d2) be metric spaces, and let y∈X2. Define
f:X1→X2 by f(x) =y for...
Let (X1,d1) and (X2,d2) be metric spaces, and let y∈X2. Define
f:X1→X2 by f(x) =y for all x∈X1. Show that f is continuous.
(TOPOLOGY)
Let X1 and X2 have the joint pdf
f(x1,x2) = 8x1x2 0<x1 <x2 <1
0....
Let X1 and X2 have the joint pdf
f(x1,x2) = 8x1x2 0<x1 <x2 <1
0. elsewhere
What are the marginal pdfs of x1 and x2?
Find the expected values of x1 and x2.
3. What is the expected value of X1X2? (Hint: Define
g(X1, X2) = X1X2 and extend the definition of expectation of
function of a random variable to two variables as follows: E[g(X1,
X2)] = ? ? g(x1, x2)f(x1, x2)dx1dx2.
4. Suppose that Y = X1/X2. What...
Let (X1, X2) have joint pdf
f(x1, x2) =
(2/9)x1x22, 0 <= x1
<= 1, 0...
Let (X1, X2) have joint pdf
f(x1, x2) =
(2/9)x1x22, 0 <= x1
<= 1, 0 <= x2 <= 3
(i) What is the distribution of Y = X1 +
X2?
(ii) What is the distribution of Y = X1 *
X2?
(iii) Find the expectation E(X1 + X2)
(iv) Find the expectation E(X1X2)
Let X = ( X1, X2, X3, ,,,, Xn ) is iid,
f(x, a, b) =...
Let X = ( X1, X2, X3, ,,,, Xn ) is iid,
f(x, a, b) = 1/ab * (x/a)^{(1-b)/b} 0 <= x <= a ,,,,, b
< 1
then,
Show the density of the statistic T = X(n) is given by
FX(n) (x) = n/ab * (x/a)^{n/(b-1}} for 0 <= x <=
a ; otherwise zero.
# using the following
P (X(n) < x ) = P (X1 < x, X2 < x, ,,,,,,,,, Xn < x
),
Then assume...