How many ordered quadruples (x1, x2, x3, x4) of positive integers satisfy the conditions x1 <...

How many ordered quadruples (x1, x2, x3, x4) of positive integers satisfy the conditions x1 < x2 < x3 < x4 < 15 and xi+1 − xi ≥ i for each i = 1, 2, 3.

Expert Solution

Colby Messinger answered 2 years ago

(a) Consider three positive integers, x1, x2, x3, which satisfy the inequality below: x1 +x2 +x3...

(a) Consider three positive integers, x1, x2, x3, which satisfy the inequality below: x1 +x2 +x3 =17. (1) Let’s assume each element in the sample space (consisting of solution vectors (x1, x2, x3) satisfying the above conditions) is equally likely to occur. For example, we have equal chances to have (x1, x2, x3) = (1, 1, 15) or (x1, x2, x3) = (1, 2, 14). What is the probability the events x1 +x2 ≤8occurs,i.e.,P(x1 +x2 ≤8|x1 +x2 +x3 =17andx1,x2,x3 ∈Z+)(Z+...

Let U = {(x1,x2,x3,x4) ∈F4 | 2x1 = x3, x1 + x4 = 0}. (a) Prove...

Let U = {(x1,x2,x3,x4) ∈F4 | 2x1 = x3, x1 + x4 = 0}. (a) Prove that U is a subspace of F4. (b) Find a basis for U and prove that dimU = 2. (c) Complete the basis for U in (b) to a basis of F4. (d) Find an explicit isomorphism T : U →F2. (e) Let T as in part (d). Find a linear map S: F4 →F2 such that S(u) = T(u) for all u ∈...

How many integer solutions are there to x1+x2+x3+x4= 100 with all of the following constraints: 10...

How many integer solutions are there to x1+x2+x3+x4= 100 with all of the following constraints: 10 ≤ x1 , 0≤ x2 < 20 , 0 ≤ x3 < 40 , 10 ≤ x4< 50 Please solve using the Principle of Inclusion/Exclusion.

The prices of inputs (x1,x2,x3,x4) are (4,1,3,2): (a) If the production function is given by f(x3,x4)...

The prices of inputs (x1,x2,x3,x4) are (4,1,3,2): (a) If the production function is given by f(x3,x4) =min⁡{x1+x2,x3+x4} what is the minimum cost of producing one unit of output? (b) If the production function is given by f(x3,x4)=x1+x2 +min⁡{x3+x4} what is the minimum cost of producing one unit of output?

4. Determine the number of integer solutions of x1 + x2 + x3 + x4 =...

4. Determine the number of integer solutions of x1 + x2 + x3 + x4 = 17, where a. xi ≥ 0, 1 ≤ i ≤ 4 b. x1, x2 ≥ 3 and x3, x4 ≥ 1 c. xi ≥ -2, 1 ≤ i ≤ 4 d. x1 , x2 , x3 > 0 and 0 < x4 ≤ 10

Let X1, X2, X3, and X4 be a random sample of observations from a population with...

Let X1, X2, X3, and X4 be a random sample of observations from a population with mean ? and variance ?2. Consider the following two point estimators of ?: b1= 0.30 X1 + 0.30 X2 + 0.30 X3 + 0.30 X4 and b2= 0.20 X1 + 0.40 X2 + 0.40 X3 + 0.20 X4 . Which of the following constraints is true? A. Var(b1)/Var(b2)=0.76 B. Var(b1)Var(b2) C. Var(b1)=Var(b2) D. Var(b1)>Var(b2)

Find a basis and the dimension of the subspace: V = {(x1, x2, x3, x4)| 2x1...

Find a basis and the dimension of the subspace: V = {(x1, x2, x3, x4)| 2x1 = x2 + x3, x2 − 2x4 = 0}

Using Y as the dependent variable and X1, X2, X3, X4 and X5 as the explanatory...

Using Y as the dependent variable and X1, X2, X3, X4 and X5 as the explanatory variables, formulate an econometric model for data that is (i) time series data (ii) cross-sectional data and (iii) panel data – (Hint: please specify the specific model here not its general form).

The values of X1, X2, X3, X4 and X5 are .1, .2, .08, .2 and .3....

The values of X1, X2, X3, X4 and X5 are .1, .2, .08, .2 and .3. Explicate the meaning of those numbers for the Z score. What are the prospects of this firm? Discuss thoroughly.

Suppose that X1, X2, X3, X4 is a simple random (independent and identically distributed) sample of...

Suppose that X1, X2, X3, X4 is a simple random (independent and identically distributed) sample of size 4 from a normal distribution with an unknown mean μ but a known variance 9. Suppose further that Y1, Y2, Y3, Y4, Y5 is another simple random sample (independent from X1, X2, X3, X4 from a normal distribution with the same mean μand variance 16. We estimate μ with U = (bar{X}+bar{Y})/2. where bar{X} = (X1 + X2 + X3 + X4)/4 bar{Y}...

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