Let X and Y have the joint pdf f(x,y)=(3/2)x^2 (1-|y|) , -1<x<1, -1<y<1.

1. Calculate Var(X).
2. Calculate Var(Y).
3. P(−X≤Y).

Starbuck's specialty coffees or teas and a breakfast sandwich per household week and indicate which unit sales and price point maximizes total revenue and contrast that with the different unit sales and price point that maximizes operating profit:

*Make sure to draw a normal distribution curve to indicate the position and range of Z.

1. Let X~N(10,9). Find P(X<8), P(X ≥ 12), P(2 ≤ X ≤ 10)

2. In each part below, find the value of c that makes the probability statement true;

1)  ∳(C) = 0.9463

2) P(IZI ≤ C) = 0.95

3) P(IZI ≤ C) = 0.99

4) P(IZI ≤ C) = 0.05

3. If X~N(80,10²), compute;

1) P(X < 100)

2) P(75 < X  < 100)

3) P(75 ≤ X)

4) P(IX-80I ≤19.6)

Consider the vector space P₂ := P₂(F) and its standard basis α = {1,x,x^2}.

1Prove that β = {x−1,x^2 −x,x^2 + x} is also a basis of P₂

2Given the map T : P2 → P2 defined by T(a + bx + cx2) = (a + b + c) + (a + 2b + c)x + (b + c)x²
compute [T]^β_α.

3 Is T invertible? Why

4 Suppose the linear map U : P2 → P2 has the matrix representation

(1 0 0

0 2 0

0 0 4)

Compute [UT]^α_α and complete the following formula
(UT)(a+bx+cx²) =

Parker Products manufactures a variety of household products. The company is considering introducing a new detergent. The company's CFO has collected the following information about the proposed product. (Note: You may or may not need to use all of this information, use only the information that is relevant.)

Estimate the project net cash flows. Make sure to put the cash flows in order: CF0 in blank 1, CF1 in Blank 2, CF2 in Blank 3, etc. Round it to a whole dollar, and do not include the $ sign.
In box 6 (last one), compute the project's NPV. Round it to a whole dollar, and do not include the $ sign.

In this question we show that we can use φ(n)/2. Let n = pq. Let x be a number so that gcd(x, n) = 1.

1. show that x^φ(n)/2 = 1 mod p and x^φ(n)/2 = 1 mod q

2. Show that this implies that and x^φ(n)/2 = 1 mod n

3. Show that if e · d = 1 mod φ(n)/2 then x^e·d = 1 mod n.

4. How can we use φ(n)/2 in the RSA?

Please explain answers if you can! Thanks!!

1. For the function f(x)=x2−36 evaluate f(x+h).

f(x+h)=

2. Let f(x)=3x+4,g(x)=9x+12, and h(x)= 9x^2+ 24x+16. evaluate the following:

a. (fg)(3)=

b. (f/g) (2)=

c. (f/g) (0)=

d.(fh)(-1)=

3. Let f(x)=2x-1, g(x)=x-3, and h(x) =2x^2-7x+3. write a formula for each of the following functions and then simplify

a. (fh) (x)=

b. (h/f) (x)=

c. (h/g) (x)=

4.Let f(x)=5−x and g(x)=x^3+3 find:

a. (f∘g)(0)=

b.(g∘f)(0)=

c. (f∘g)(x)=

d. (g∘f)(x)=

5. Let f(x)=x^2+5x and g(x)=4x+5 find:

a. (f∘g)(x)=

b. (g∘f)(x)=

c. (f∘g)(0)=

d. (g∘f)(0)=

6. Let f(x)=x^2 and g(x)=x−5 find:

a. (f∘g)(x)=

b. (g∘f)(x)=

c. (f∘g)(5)=

d. (g∘f)(5)=

A 5th filter is described by the difference equation: 2y(n)=2 x(n)+7 x(n-1)+3 x(n-2)-8 x(n-3)+ x(n-4)-8 x(n-5)+7 y(n-1)-3 y(n-2)+5y(n-3)- y(n-4) Determine the frequency response. Plot the magnitude and the phase response of this filter. Consider the plot -π≤w≤π for 501 points. Describe the magnitude response (Low pass filter, High Pass filter, etc.) Determine the system stability. Determine the impulse response h(n). You may set the period to -100≤n≤100 Determine the unit step response for -100≤n≤100 . (Matlab)

Data are collected on 15 individuals and two models are considered:

Model 1: model with 2 predictors (A and B), SStotal = 360, SST = 200

Model 2: model with 4 predictors (A, B, C and D), SStotal = 360, SST = 270

1. The standard error of the complete model is equal to _______

2. The number of predictors to test to evaluate whether the restricted model is sufficient is equal to _______

3. The partial F-test in 2. is equal to _______

4. The critical value (rejection region) for the partial F-test is equal to _______

5. The adjusted coefficient of determination for the complete model is equal to ______ %