Exercise 4. (The Pythagorean theorem) Show that in a right triangle the square of the length of the hypotenuse is equal to the sum of the squares of the length of the sides. Next let us define the angle defined by 3 points p, o, q as (p – 0,9 – o) Zpoq := angle(p – 0,q – o) = cos ||p – o|| ||g – o||

Q: Exercise 12. Let X, Y, Z N(0,1) be independent. Determine the distribution, expectation and variance…

A: “Since you have posted multiple questions, we will provide the solution only to the first question…

Q: Exercise 11. Compute the area of the pseudo-sphere, i.e. the surface of revolution obtained by…

A: Area= ∫2πrds=2π∫r.dr/sin(ϕ)∫2πrds=2π∫r.dr/sin(ϕ) Due to tangent property: r/sin(ϕ)=a,r/sin(ϕ)=a, and…

Q: Exercise 5. Show that there exists at most one closed geodesic on a cylinder with negative…

A: We will assume S ⊂ R be a surface homeomorphic to a cylinder and with Gaussian curvature K < 0.…

Q: Exercise 2 Let X be a topological space. TALING TI 2) Suppose X = Uier B₁, with each B; C X. If A CX…

A:

Q: Exercise 2 Let 1 B = 3 C = These are bases for R? (no need to prove this). Find the change of…

A:

Q: 3) Suppose X = Uier Ui, with each U₁ an open subset of X. If for each i E I, the subset U₁ is a…

A: Kindly refer Step 2 for the solution.

Q: olve by variation of parameters the following differential equations 1. "+ 9y = tan 3r

A: Exercise 3 1. 1)

Q: Exercise 1) Assume that X is a Hausdorff space. If D C X is a dense subset of X. If the point xD is…

A:

Q: 11. Show that a finite integral domain is a division ring. 12. If x= x for all x in a ring R, show…

A:

Q: Exercise 3. Compute the Christoffel symbols of a surface of revolution.

A: The Christoffel symbols are an array of numbers describing a metric connection.We need to compute…

Q: Is Shur's arm lemma true for nonconvex arcs?

A: We need to check Exercise 8 which asks "Is Shur's arm lemma true for nonconvex arcs?". For getting…

Q: Exercise 3. Show that the Gaussian curvature of a surface which is home- omorphic to the torus must…

A:

Q: Exercise 3. Suppose V is finite-dimensional and t,., tm is a linearly independent list in V'. Prove…

A: We use Fundamental theorem of Linear maps to prove this result.

Q: Exercise 6 Is the set of vectors V = {i² + 2t – 1,3t + 5, -t² + t + 6} linearly dependent?

A:

Q: Exercise 7 Assume that you have a set of vectors D = {ü1, ö2, ö3, ... ön} and the linear combination…

A: We have to prove given statement:

Q: Exercise 2. Show that the total geodesic curvature of a simple closed planar curve is 27.

A: We will use the Tangent-Normal frame to prove the given result.

Q: Exercise 3. For an i.i.d. sample of n = 2m + 1 observations of a U(0, 1) random variable, show that…

A: Given an iid sample of n=2m+1 observations of a U0,1 random variable.

Q: Suppose that F(x) = f(g(x)), where ƒ(2) = 3, g(2) = 5, g' (2) = 4, ƒ' (2) = -2, and ƒ' (5) = 11.…

A:

Contingency Table

A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.

Binomial Distribution

Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.