Corollary of theorem - Introduction:
In math a corollary normally follows a theorem. The exercise of the expression corollary, slightly than proposal or theorem, is basically subjective. Proposal B is corollaries of proposal A if B can readily be assume from A, but the meaning of gladly varies depending on the author and context. The importance of the corollary is frequently measured secondary to that of the primary theorem; B is suspect to be termed a corollary if its math consequences are as important as those of A. Occasionally a corollary has a proof that explicate the derivation; on occasion the derivation is considered to be self-evident.
Corollary of some Theorems:
Corollary of Theorem
Let Y ~ B (n, `pi` ), where `pi` is called the population proportion, n is the sample size and Y is the number of “success”s in the sample. Then the sample proportion is p = `Y/n` . Given these,
`lim_(x->oo)P(|y/n - pi| <= epsi)= lim_(x->oo)P(|p-pi|<=epsi) =1`
This states that as the sample size increases the sample proportion converges in probability to the population proportion, that is, p->`pi` .
Theorem
Strong Law of Large Numbers or almost sure convergence
Let X1, X2,.. Xn be a random sample from a population (of X’s) with mean E(X) = mX and a finite variance, Var(X) = `sigma^2_x < oo` . Let `bar(X) = 1/n sum_(i=1)^n X_i` . Then, for any positive real number e,
`P(lim_(n->oo) |bar(X)- mu_x| <= epsi =1 or p(lim_(n->oo)) | bar(X) - mu_x|>=epsi) = 0`
Angle Sum Theorem:
The sum of the angle measures in any triangle is 180°
PROVE IT!
x + x + y = 180° (Triangle)
y + y + y = 180° (Line)
y = x (Alternate Interior Angles)
Corollary to the Angle Sum Theorem:
If ∆ABC is any triangle, then an exterior angle of ABC has the same measure as the sum of the measures of the two nonadjacent interior angles.
PROVE IT!
m `angle` 1 + m `angle` 2 = m `angle` 3
I have recently faced lot of problem while learning equation for profit, But thank to online resources of math which helped me to learn myself easily on net.
Altitude Similarity Theorem
The altitude to the hypotenuse of a right triangle divides the triangle into two triangles that are similar to the original triangle and to each other.
`Delta` ABC ~ `Delta` ACD ~ `Delta ` CBD
Corollary 1
The length of the altitude to the hypotenuse of a right triangle is the geometric mean of the lengths of the segments of the hypotenuse.
`(AD)/(CD) = (CD)/(DB) CD = sqrt(AD(DB)) `
Corollary 2
The altitude to the hypotenuse of a right triangle separates the hypotenuse so that the length of each leg of the triangle is the geometric mean of the length of the adjacent hypotenuse segment and the length of the hypotenuse.
`(AD)/(AC) = (AC)/(AB') , (BD)/(CB) = (CB)/(AB)`
In math a corollary normally follows a theorem. The exercise of the expression corollary, slightly than proposal or theorem, is basically subjective. Proposal B is corollaries of proposal A if B can readily be assume from A, but the meaning of gladly varies depending on the author and context. The importance of the corollary is frequently measured secondary to that of the primary theorem; B is suspect to be termed a corollary if its math consequences are as important as those of A. Occasionally a corollary has a proof that explicate the derivation; on occasion the derivation is considered to be self-evident.
Corollary of some Theorems:
Corollary of Theorem
Let Y ~ B (n, `pi` ), where `pi` is called the population proportion, n is the sample size and Y is the number of “success”s in the sample. Then the sample proportion is p = `Y/n` . Given these,
`lim_(x->oo)P(|y/n - pi| <= epsi)= lim_(x->oo)P(|p-pi|<=epsi) =1`
This states that as the sample size increases the sample proportion converges in probability to the population proportion, that is, p->`pi` .
Theorem
Strong Law of Large Numbers or almost sure convergence
Let X1, X2,.. Xn be a random sample from a population (of X’s) with mean E(X) = mX and a finite variance, Var(X) = `sigma^2_x < oo` . Let `bar(X) = 1/n sum_(i=1)^n X_i` . Then, for any positive real number e,
`P(lim_(n->oo) |bar(X)- mu_x| <= epsi =1 or p(lim_(n->oo)) | bar(X) - mu_x|>=epsi) = 0`
Angle Sum Theorem:
The sum of the angle measures in any triangle is 180°
PROVE IT!
x + x + y = 180° (Triangle)
y + y + y = 180° (Line)
y = x (Alternate Interior Angles)
Corollary to the Angle Sum Theorem:
If ∆ABC is any triangle, then an exterior angle of ABC has the same measure as the sum of the measures of the two nonadjacent interior angles.
PROVE IT!
m `angle` 1 + m `angle` 2 = m `angle` 3
I have recently faced lot of problem while learning equation for profit, But thank to online resources of math which helped me to learn myself easily on net.
Altitude Similarity Theorem
The altitude to the hypotenuse of a right triangle divides the triangle into two triangles that are similar to the original triangle and to each other.
`Delta` ABC ~ `Delta` ACD ~ `Delta ` CBD
Corollary 1
The length of the altitude to the hypotenuse of a right triangle is the geometric mean of the lengths of the segments of the hypotenuse.
`(AD)/(CD) = (CD)/(DB) CD = sqrt(AD(DB)) `
Corollary 2
The altitude to the hypotenuse of a right triangle separates the hypotenuse so that the length of each leg of the triangle is the geometric mean of the length of the adjacent hypotenuse segment and the length of the hypotenuse.
`(AD)/(AC) = (AC)/(AB') , (BD)/(CB) = (CB)/(AB)`
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