To derive the linear pricing equations, you impose the no-arbitrage condition that the current price of each traded asset must equal the sum of its state-contingent payoffs weighted by the corresponding state prices. This produces one linear equation per traded asset, with the state prices acting as the unknowns.
Conditions: One-period model with a finite set of states.; Prices of all traded assets are known today.; State-contingent payoffs for each asset are known.; The unknowns are the state prices (Arrow-Debreu security prices).
To derive the linear pricing equations, you impose the no-arbitrage condition that the current price of each traded asset must equal the sum of its state-contingent payoffs weighted by the corresponding state prices. This produces one linear equation per traded asset, with the state prices acting as the unknowns.
Conditions: One-period model with a finite set of states.; Prices of all traded assets are known today.; State-contingent payoffs for each asset are known.; The unknowns are the state prices (Arrow-Debreu security prices).
To convert a percentage into a fraction, place the numerical value of the percentage over 100. The number before the percent sign becomes the numerator, and the denominator is fixed at 100.
Conditions: The starting value is in percent form.; The denominator must be 100.
To convert a percentage into a fraction, place the numerical value of the percentage over 100. The number before the percent sign becomes the numerator, and the denominator is fixed at 100.
Conditions: The starting value is in percent form.; The denominator must be 100.
To find the left-hand limit as x approaches a value c (denoted limx→c−f(x)), observe the behavior of the function's curve strictly for input values less than c, moving towards c. If the y-values grow without bound in either the positive or negative direction, the finite real limit does not exist.
Conditions: The function has a vertical asymptote or discontinuity at x=c.; You are evaluating the limit from the left side (inputs strictly less than c).
To find the left-hand limit as x approaches a value c (denoted limx→c−f(x)), observe the behavior of the function's curve strictly for input values less than c, moving towards c. If the y-values grow without bound in either the positive or negative direction, the finite real limit does not exist.
Conditions: The function has a vertical asymptote or discontinuity at x=c.; You are evaluating the limit from the left side (inputs strictly less than c).
To scale an expected value, calculate the expectation for a single unit and then multiply it by the total number of units. This relies on the linearity of expectation, which holds regardless of whether the units are independent, provided the individual expectations are defined.
Conditions: The expectation for a single unit is known or calculable.; The number of units is fixed and finite.
To scale an expected value, calculate the expectation for a single unit and then multiply it by the total number of units. This relies on the linearity of expectation, which holds regardless of whether the units are independent, provided the individual expectations are defined.
Conditions: The expectation for a single unit is known or calculable.; The number of units is fixed and finite.
Min-Max normalization is a method that linearly transforms raw data into the interval [0, 1]. The formula is X′=max−minX−min, where X is the original value, and min and max are the minimum and maximum values of that feature column, respectively.
Conditions: The dataset has clear min and max values.; max is not equal to min
Min-Max normalization is a method that linearly transforms raw data into the interval [0, 1]. The formula is X′=max−minX−min, where X is the original value, and min and max are the minimum and maximum values of that feature column, respectively.
Conditions: The dataset has clear min and max values.; max is not equal to min
The x/ln(x) approximation yields an estimated count of 3.1 trillion primes below 100 trillion. Compared to the actual count of 3.2 trillion, this is over 99.99% accurate.
Conditions: x=100,000,000,000,000.; The formula used is #primes(x) ~ x/ln(x).
The x/ln(x) approximation yields an estimated count of 3.1 trillion primes below 100 trillion. Compared to the actual count of 3.2 trillion, this is over 99.99% accurate.
Conditions: x=100,000,000,000,000.; The formula used is #primes(x) ~ x/ln(x).