Vectors of Random Variables
... Random vectors are discrete, continuous or neither. The definition is very similar to the one dimensional ~ is discrete if there exists a finite or countable set K s.t. P (X ~ is continuous if ~ ∈ K) = 1 and X case: X ~ = (x1 , . . . , xn )) = 0 for all (x1 , . . . , xn ). For example, the voltage m ...
... Random vectors are discrete, continuous or neither. The definition is very similar to the one dimensional ~ is discrete if there exists a finite or countable set K s.t. P (X ~ is continuous if ~ ∈ K) = 1 and X case: X ~ = (x1 , . . . , xn )) = 0 for all (x1 , . . . , xn ). For example, the voltage m ...
1.3 Binomial Coefficients
... Further the second half of each row is the reverse of the first half. The array of numbers called Pascal’s Triangle emphasizes that symmetry by rearranging the rows of the table so that they line up at their centers. We show this array in Table 2. When we write down Pascal’s triangle, we leave out th ...
... Further the second half of each row is the reverse of the first half. The array of numbers called Pascal’s Triangle emphasizes that symmetry by rearranging the rows of the table so that they line up at their centers. We show this array in Table 2. When we write down Pascal’s triangle, we leave out th ...
MATH 645 - HOMEWORK 1: Exercise 1.1.3. Show that
... Exercise 1.2.1. Let H be a Hilbert space and suppose f and g are linearly independent vectors in H with kf k = kgk = 1. Show that ktf + (1 − t)gk < 1 for 0 < t < 1. What does this say about {h ∈ H : khk ≤ 1}. Proof. By the proof of the triangle inequality and the result on equality from the Cauchy-S ...
... Exercise 1.2.1. Let H be a Hilbert space and suppose f and g are linearly independent vectors in H with kf k = kgk = 1. Show that ktf + (1 − t)gk < 1 for 0 < t < 1. What does this say about {h ∈ H : khk ≤ 1}. Proof. By the proof of the triangle inequality and the result on equality from the Cauchy-S ...
Fundamental theorem of calculus
The fundamental theorem of calculus is a theorem that links the concept of the derivative of a function with the concept of the function's integral.The first part of the theorem, sometimes called the first fundamental theorem of calculus, is that the definite integration of a function is related to its antiderivative, and can be reversed by differentiation. This part of the theorem is also important because it guarantees the existence of antiderivatives for continuous functions.The second part of the theorem, sometimes called the second fundamental theorem of calculus, is that the definite integral of a function can be computed by using any one of its infinitely-many antiderivatives. This part of the theorem has key practical applications because it markedly simplifies the computation of definite integrals.