1. Use mathematical induction to show that, ∀n ≥ 3,
2n2 + 1 ≥ 5n
2....
1. Use mathematical induction to show that, ∀n ≥ 3,
2n2 + 1 ≥ 5n
2. Letting s1 = 0, find a recursive formula for the
sequence 0, 1, 3, 7, 15,...
3. Evaluate. (a) 55mod 7. (b) −101 div 3.
4. Prove that the sum of two consecutive odd integers is
divisible by 4
5. Show that if a|b then −a|b.
6. Prove or disprove: For any integers a,b, c, if a ∤ b and b ∤
c, then...
Please answer Problems 1 and 2 thoroughly.
Problem 1: Let X be a set. Define a...
Please answer Problems 1 and 2 thoroughly.
Problem 1: Let X be a set. Define a partial ordering ≤ on P(X)
by A ≤ B if and only if A ⊆ B. We stated the following two facts in
class. In this exercise you are asked to give a formal proof of
each:
(a) (1 point) If A, B ∈ P(X), then sup{A, B} exists, and sup{A,
B} = A ∪ B.
(b) (1 point) If A, B ∈ P(X),...
Assume a population of 1, 2, and 12. Assume that samples of
size n equals 2...
Assume a population of 1, 2, and 12. Assume that samples of
size n equals 2 are randomly selected with replacement from the
population. Listed below are the nine different samples. Complete
parts a through d below. 1,1 1,2 1,12 2,1 2,2 2,12 12,1
12,2 12,12
a. Find the value of the population standard deviation
b. Find the standard deviation of each of the nine samples, then
summarise the sampling distribution of the standard deviations in
the format of a...
Let the population have N=7 units, with {(unit,value)} =
{(A,-1),(B,+1),(C,-2),(D,+3),(E,-4),(F,+5),(G,-6)}. The design is
as follows: first...
Let the population have N=7 units, with {(unit,value)} =
{(A,-1),(B,+1),(C,-2),(D,+3),(E,-4),(F,+5),(G,-6)}. The design is
as follows: first choose A or B at random; if A then choose from
{C,D} at random, if B then choose from {E,F,G} at random. 1) Find
the first-order inclusion probabilities (note that the sample size
n is fixed at 2).
Verify (show numerically for this example) that the
Horvitz-Thompson estimator is unbiased for the population total.
(Hint: find the probability of each sample and the value...
1) State the main difference between an ODE and a PDE?
2) Name two of the...
1) State the main difference between an ODE and a PDE?
2) Name two of the three archetypal PDEs?
3) Write the equation used to compute the Wronskian for two
differentiable
functions, y1 and y2.
4) What can you conclude about two differentiable functions, y1 and
y2, if their
Wronskian is nonzero?
5) (2 pts) If two functions, y1 and y2, solve a 2nd order DE, what
does the Principle of
Superposition guarantee?
6) (8 pts, 4 pts each) State...