Write out a table of discrete logarithms modulo 17 with respect to the primitive root3. If m is a positive integer, the integer a is a quadratic residue of m if gcd ( a , m ) = 1 and the congruence x 2 ≡ a ( mod m ) has a solution. In other words, a quadratic residue of m is an integer relatively prime to m that is a perfect square modulo m . If a is not a quadratic residue of m and gcd ( a , m ) = 1 , we say that it is a quadratic nonresidue of m . For example, 2 is a quadratic residue of 7 because gcd ( 2 , 7 ) = 1 and 3 2 ≡ 2 ( mod 7 ) and 3 is a quadratic nonresidue of 7 because gcd ( 3 , 7 ) = 1 and x 2 ≡ 3 ( mod 7 ) has no solution.
Write out a table of discrete logarithms modulo 17 with respect to the primitive root3. If m is a positive integer, the integer a is a quadratic residue of m if gcd ( a , m ) = 1 and the congruence x 2 ≡ a ( mod m ) has a solution. In other words, a quadratic residue of m is an integer relatively prime to m that is a perfect square modulo m . If a is not a quadratic residue of m and gcd ( a , m ) = 1 , we say that it is a quadratic nonresidue of m . For example, 2 is a quadratic residue of 7 because gcd ( 2 , 7 ) = 1 and 3 2 ≡ 2 ( mod 7 ) and 3 is a quadratic nonresidue of 7 because gcd ( 3 , 7 ) = 1 and x 2 ≡ 3 ( mod 7 ) has no solution.
Write out a table of discrete logarithms modulo 17 with respect to the primitive root3.
If m is a positive integer, the integer a is a quadratic residue of m if
gcd
(
a
,
m
)
=
1
and the congruence
x
2
≡
a
(
mod
m
)
has a solution. In other words, a quadratic residue of m is an integer relatively prime to m that is a perfect square modulo m. If a is not a quadratic residue of m and
gcd
(
a
,
m
)
=
1
, we say that it is a quadratic nonresidue of m. For example, 2 is a quadratic residue of 7 because
gcd
(
2
,
7
)
=
1
and
3
2
≡
2
(
mod 7
)
and 3 is a quadratic nonresidue of 7 because
gcd
(
3
,
7
)
=
1
and
x
2
≡
3
(
mod 7
)
has no solution.
2. Consider the negative binomial distribution with parameters r,p and having pmf
nb(x;r,p) =
Ꮖ
(* + r − ¹) p*(1 − p)²
p'(1-p) x = 0, 1, 2, 3, … ….
(a) Supposer 2, then show that
T-1
p =
X+r−1
is an unbiased estimator for p. (Hint: write out E(p), then cancel out x+r −1 inside
the sum).
(b) A reporter wishing to interview five individuals who support a certain candidate (for
presidency?) begins asking people whether they support (S) or not support (F) the can-
didate.
If they observe the following sequence of responses SFFSfffffffffffSSS, esti-
mate p the true proportion of people who support the candidate.
How does the estimate change if the following sequence of responses were observed
ssssfffffffffffffs.
Does it matter to the estimate when the first four S's appear in the sequence of responses?
Let A =
23
231
3 54
Find a basis for Row A.
Find a basis for Col A.
Find a basis for Nul A.
7
in Nul A? Why or why not?
2
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