Question

Given that A_{1} = B_{1} minus B_{2,}
A_{2} = B_{2} minus B_{3}, and
A_{3} = B_{3} minus B_{1,} Find the joint
p.m.f. (probability mass function) of A_{1} and
A_{2}, where A_{i} ~ Ber(p) for all random
variables *i* in {1,2,3}

Answer #1

Given that A1 = B1 minus
B2,A2 =
B2 minus B3, and
A3 = B3 minus
B1, Find the joint p.m.f.
(probability mass function) of A1 and A2,
where Bi ~ Ber(p) for all
random variables i in {1,2,3}

Consider the ring R = Z ∞ = {(a1, a2, a3, · · ·) : ai ∈ Z for
all i}. It turns out that R forms a ring under the operations (a1,
a2, a3, · · ·) + (b1, b2, b3, · · ·) = (a1 + b1, a2 + b2, a3 + b3,
· · ·), (a1, a2, a3, · · ·) · (b1, b2, b3, · · ·) = (a1 · b1, a2 ·
b2, a3 ·...

If the determinant of the 3x3 matrix [2c1 -2c2 2c3; a1+6c1
-a2-6c2 a3+6c3; -b1 b2 -b3] is -16, what is the determinant of the
3x3 matrix [a1 b1 c1; a2 b2 c2; a3 b3 c3]?

Consider the ring R = Z∞ = {(a1,a2,a3,···) : ai ∈ Z for all
i}.
It turns out that R forms a ring under the operations:
(a1,a2,a3,···)+(b1,b2,b3,···)=(a1 +b1,a2 +b2,a3 +b3,···),
(a1,a2,a3,···)·(b1,b2,b3,···)=(a1 ·b1,a2 ·b2,a3 ·b3,···)
Let I = {(a1,a2,a3,···) ∈ Z∞ : all but finitely many ai are 0}.
You may use without proof the fact that I forms an ideal of R.
a) Is I principal in R? Prove your claim.
b) Is I prime in R? Prove your claim....

Consider the ring R = Z∞ = {(a1,a2,a3,···) : ai ∈ Z for all
i}.
It turns out that R forms a ring under the operations:
(a1,a2,a3,···)+(b1,b2,b3,···)=(a1 +b1,a2 +b2,a3 +b3,···),
(a1,a2,a3,···)·(b1,b2,b3,···)=(a1 ·b1,a2 ·b2,a3 ·b3,···)
Let I = {(a1,a2,a3,···) ∈ Z∞ : all but finitely many ai are 0}.
You may use without proof the fact that I forms an ideal of R.
a) Is I principal in R? Prove your claim.
b) Is I prime in R? Prove your claim....

consider a sample space defined by events a1, a2, b1 and b2
where a1 and a2 are complements .given p(a1)=0.2 p(b1/a1) = 0.5 and
p(b1/a2) =0.7 what is the probability of p (a1/b1)
P(A1/B1)=
round to the 3rd decimal

Calculate P(A1 I B2) given the following table.
A1 A2
B1 0.4 0.2
B2 0.1 0.3

Write a function of the form function [x1pts,x2pts] =
unif_over_rect(a1,b1,a2,b2,n) which provides the coordinates
(x1pts(i),x2pts(i)), 1 ≤ _i ≤ _n, of n random darts (more
precisely, realizations of random darts) thrown at the rectangle a1
≤ x1 ≤ b1, a2 ≤ x2 ≤ b2. (The lower left corner of the rectangle is
a1,a2; the upper right corner of the rectangle is b1,b2.) You may
assume that the darts are drawn from the bivariate uniform
distribution over the rectangle and hence...

Given the following *joint probability distribution*, P(A,B),
for A and B,
a1
a2
b1
0.37
0.16
b2
0.23
0.24
Calculate the marginal probability distribution,
P(B).
Calculate the conditional probability distribution,
P(A|B).

Explain why the set of vectors given in matrix form do not span
R3 : a1 b1 2a1
-3b1
a2 b2 2a2 -3b2
a3 b3 2a3 -3b3

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