Shakespeare used the word probable, which then meant trust worthy (2333)
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Some gamblers must have used probability theory before the 16th century, it's just not documented
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Insurance is based on guarantees for independent events, P(ABC...) = P(A)*P(B)*P(C). …
The probability of all events happening is very small. As long as the insurance company sells enough insurance policies, the company can make steady profits with almost no risk.
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Geometric mean means (x1*x2...*xn)**(1/n) The geometric mean must be less than or equal to (the equal sign when all variables are the same) the arithmetic mean
For example, for ten consecutive years of investment, the average annual gross return (return rate + 1) in these years should be a geometric average!
If the yield is 20% in the first nine years and -100% in the last year, then it's all lost after ten years.
Using the summed average, the average rate of return is 18%, which obviously does not reflect the real situation.
The geometric mean is 0
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D (x) = E ((x-ux) ** 2)
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When sampling is used to estimate the population, the expectation of the mean of the sampling set is the same as the mean of the population, but the expectation of the variance of the sampling set is equal to (n-1)/n times the population variance.
Therefore, when calculating the variance of the sampling set, divide by (n-1), not n, to obtain an unbiased estimate of the population.
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Returns need high mean, low variance (meaning low risk)
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Covariance, when there are two random variables, reflects the trend correlation of the two variables, the same trend covariance is positive, otherwise it is negative
The correlation coefficient (correlation) is between (-1, 1)
Correlation coefficient = cov(x,y)/(sx*xy) product of covariance divided by standard deviation
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fat tailed distribution is a wider Gaussian distribution
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present （discounted） value
The bank promises to give me a dollar in a year (the bank will not break its word, and the bank will definitely give me at the end of the year, because the bank can use my dollar to invest), then I want the bank to give this money to me now Me, how much will the bank give me?
1/(1+r) r is the annual interest rate
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consol (consolidated debt public debt) perpetuity (perpetual annuity)
If you are given a coupon of one dollar per year for three years, what is the present value?
1/(1+r） + 1/(1+r)**2 + 1/(1+r)**3
If it is exchanged indefinitely, with c dollars per year, then the present value is c/r dollars
It can be seen that pv is inversely proportional to the interest rate
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growing consol
It will give you c dollars after one year, c*(1+g) dollars after another year, c*(1+g)**2 dollars after another year, and so on, exchange forever
#Note, g is less than r, because this is common sense
pv = c/(1+r) + [c*(1+g) / (1+r)**2] + [c*(1+g)**2 / (1+r)**3] +........
= c/(r-g)
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annuity annuity
Give you c yuan per year for n years, then the present value is C*[1 - (1 / 1-r)**n]/r
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utility function
diminishing marginal utility
The Ux curve of people's satisfaction with the income x yuan is as follows, it can be seen that there is a phenomenon of diminishing marginal utility
A little less money for the rich has no effect, and a little more for the poor makes a lot happier.
You can evenly divide the months you have money into the months you don't have money. . . Because what you want is a year's average of happy E[u(x)] max
The theory of utility expectation maximization (people want to maximize utility expectation) is not always true, but it is a core concept.