Keep your eyes and ears open as you read or listen to the news this week. Find/discover an example of statistics & probability in the news. Was it explained well, or poorly? What is the context? Do you think anything about the article is misleading? Can you make any inferences based on it?

Do you think that the statistics in the article would convince the reader that we are all facing a serious problem with our environment? Explain why or why not by referencing examples from the article?

Solve the initial value problem.

(x^2 * D^2 +xD - 4i) * y = x^3, y(1) = -4/5, y'(1) = 93/5

let A be a real matrix of size M*N.,assume that nullity (A^T* A)=r. Find the range of values that 'r' can take using values of 'M' and 'N'. also find the nullity (A^T)

Problem 3-12

Suppose that you sell short 250 shares of Xtel, currently selling for $100 per share, and give your broker $15,000 to establish your margin account.

a. If you earn no interest on the funds in your margin account, what will be your rate of return after one year if Xtel stock is selling at: (i) $115; (ii) $100; (iii) $95? Assume that Xtel pays no dividends. (Leave no cells blank - be certain to enter "0" wherever required. Negative values should be indicated by a minus sign. Round your answers to 2 decimal places.)

b. If the maintenance margin is 25%, how high can Xtel’s price rise before you get a margin call? (Round your answer to 2 decimal places.)

c. Redo parts (a) and (b), but now assume that Xtel also has paid a year-end dividend of $1 per share. The prices in part (a) should be interpreted as ex-dividend, that is, prices after the dividend has been paid. (Negative values should be indicated by a minus sign. Round your answers to 2 decimal places.)

State the linearity of the equation: x (y^2 + 2z) p − y (x^2 + 2z) q = (x^2 −y^2) z. Find the general solution, and then the particular solution that passes through the line: x + y = 0, z = 2.
Linearity is the category to which it belongs among: linear, quasi-linear or non-linear.

1. Find the Laplace transform of each of the following functions: (a). f(t) = t , (b). f(t) = t²  , (c) f(t) = tⁿ  where n is a positive integer

Laplace transform of the given function

2. . f(t) = sin bt

3.   f(t) = e^at sin bt

By induction:

1. Prove that Σⁿ_i=1(2i − 1) = n²

2. Prove thatΣⁿ_i=1 i² = n(n+1)(2n+1) / 6 .

Describe the set of all unit tight frames with exactly five vectors, up to PRR-equivalence.

1.2.2*. Find an example to show that the phrase "finitely many" is necessary in the statement of Lemma 1.2.3 (iii).

Lemma 1.2.3.

(i) Rn and empty set are open in Rn
(ii) The union ofo pen subset of Rn is open.

(iii) The intersect of finitely many open subsets Rn is open.

Explain what it is a neutral theorem in Euclidean geometry.
State & prove both: the theorem on construction of parallel lines and its converse. Which one of them is neutral?

Let V be a vector space, and suppose that U and W are both subspaces of V. Show that U ∩W := {v | v ∈ U and v ∈ W} is a subspace of V.

3. Every element of Sn can be written as a product of disjoint cycles. If σ = (i1 i2)(j1 j2)(k1 k2 k3 k4),
with the cycles disjoint, we say that σ has cyclic structure ( )( )( ).
(a) Find all possible cyclic structures in S7. Hint: There are 15.
(b) Using part (a), find all possible orders in S7.
(c) Find all possible orders in A7.

1. Poutine Cheez Company has yearly sales of $550,000 and an average collection period of 35 days. A factoring company is offering a 35-day receivables loan equal to 85% of the accounts receivable at 9% along with a commission fee of .45% of the receivables. The firm estimates that by taking the offer, it could save $300 in collection costs and a full half of one percent in bad debt costs, as a percentage of sales. What is the annual cost (in percent) of the arrangement to Poutine Cheez?

Suppose you Do not know anything about Extended Euclidean Algorithm. How to find t(x) and s(x) that satisfy the greatese common divisor of f(x) and g(x) equals to f(x)t(x)+g(x)s(x) in Q(x). You can give me an example(polynomials) if you want. Thank you!