PROJECTIVE AND AFFINE GEOMETRY

¿ which is the affine application that transforms each line into another with its same direction and leaves at least two invariant points ?

Thank you for your explanations

List two practical reasons for using a small N design. Why is it usually important to return to baseline conditions in a small N design? Under what conditions would it not be desirable to return to baseline?

An individual has three umbrellas, some at his office, and some at home. If he is leaving home in the morning (or leaving work at night) and it is raining, he will take an umbrella, if there is one there. Otherwise, he gets wet. Assume that, independent of the past, it rains on each trip with probability 0.2. To formulate a Markov chain, let Xn be the number of umbrellas at his current location “before” he starts his n-th trip. Note that “current location” can either be home or office, depending on whether the trip is from home to office or vice versa. Find the transition probabilities of this Markov chain

True or false

A) you are trying to estimate a parameter of interest, you have two estimators with the same variance but the first estimator is an un-biased estimator and the second one is a biased estimator ,. In this case you should choose the un/ biased estimator

True?. False?
Please Explain

B.) you are trying to estimate Q1 ( the 25th percentile) for the anual household income in the us based on a sample of households earnings can you suggest an estimator for Q1!? Please explain why you believe this is a good estimator

What is the number of ordered sequences of length k where each digit is
taken from a set of size n?

What is the number of ordered sequences of length k where each digit is
taken from a set of size n without repetition?

What is the number of subsets of size k of a set of size n?

How do you find all solutions to: a^2+2b^2=c^2 ?

Discuss the determinants of future dividends and growth rate in dividends, and the sensitivity of the stock price to estimates of those two factors.

State and prove spectral mapping theorem

1.Suppose you have a function such that the domain of is −4≤x≤2

and the range of is −1≤y≤6.

a. What is the domain and range of the transformation

f(2(x+3))?

b. What is the domain and range of the transformation

2f(x)−3?

c. How do you know your answers are correct?

d. What can we say about how transformations affect the domain and range of a function?

2. Suppose a local vendor charges $2 per hot dog and that the number of hot dogs sold per hour x is given by x(t)=−4t^2+20t+92, where t is the number of hours since 10 AM, 0≤t≤4

a. Find an expression for the revenue per hour R as a function of x.

b. Find and simplify (R∘x)(t). What does this represent?

c. What is the revenue per hour at noon?

d. If the price were raised to $3 per hot dog with no change in the x(t) equation, which hour would produce the most revenue? Why?

e. If the price were dropped to $1 per hot dog, but that price drop caused the number of sales to increase according to the function x(t)=−9t^2+22t+138, would the vendor make more money at the original $2 price, or at the $1 price?

3. Danielle makes the claim that when the polynomial x^2−3x−10 is divided by x−5, the remainder is 0. Use what you have learned about dividing polynomials to either verify that Danielle is correct or prove that she is incorrect. What arguments would you use to support your claim? Are there any other arguments? Justify your answers.

1d. Explain the three criteria we use to evaluate solution concepts, & use them to compare Dominant Strategy Equilibrium with Nash Equilibrium?

Consider the equation
y′′ − 5y′ + 6y =e^3tcos(2t) + e^2t(3t+4)sint

(a) Determine a suitable form for yp if the method of undetermined coefficients is used

- Show steps for getting A, B, etc.

- if possible, split up the right hand side and handle each value seperately, aka y′′ − 5y′ + 6y =e^3tcos(2t),

then y′′ − 5y′ + 6y =e^2t(3t+4)sint, then adding together for final solution.

(b) Find the general solution of the equation.

R simulation:
Let X1, . . . , Xn be i.i.d. random variables from a uniform distribution on [0, 2]. Generate
and plot 10 paths of sample means from n = 1 to n = 40 in one figure for each case. Give
some comments to empirically check the Law of Large Numbers.

(a) When n is large,
X1 + · · · + Xn/n  converges to E[Xi].
(b) When n is large,
X1^2+ · · · + Xn^2/n converges to E[Xi^2 ]

T or F

1) Any N vectors spanning R^n are linearly independent

2)R5 has 7 linearly independent vectors

3) If a set of vectors with n elements is linearly dependent, then a set with n - 1 elements is also linearly dependent

4) There exists a Linear Function T:R^n -> R^n such that the range and the kernel of T are equal.

5) If a vector space has a dimension of n, then a basis for the vector space will contain n vectors

6) L: R^6 -> R^7 is one-to-one, then the range of L has a dimension of 7

7) The number of leading terms in ref(A) is equal to the dimension of the row space of matrix A

8) If a set of 4 vectors is linearly independent, then if you remove 1 vector, the set will still be linearly independent

Show that a set S has infinite elements if and only if it has a subset U such that (1) U does not equal to S and (2) U and S have the same cardinality.