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A little algebra shows that the distance between P and M (which is the same as the orthogonal distance between P and the line L) (¯) is equal to the standard deviation of the vector (x 1, x 2, x 3), multiplied by the square root of the number of dimensions of the vector (3 in this case).
The simplest estimators for population mean and population variance are simply the mean and variance of the sample, the sample mean and (uncorrected) sample variance – these are consistent estimators (they converge to the value of the whole population as the number of samples increases) but can be improved.
To determine the value (), note that we rotated the plane so that the line x+y = z now runs vertically with x-intercept equal to c. So c is just the distance from the origin to the line x + y = z along the perpendicular bisector, which meets the line at its nearest point to the origin, in this case ( z / 2 , z / 2 ) {\displaystyle (z/2,z/2)\,} .
Suppose X is a coin flip with the probability of heads being h. Suppose that when X = heads then Y is drawn from a normal distribution with mean μ h and standard deviation σ h, and that when X = tails then Y is drawn from normal distribution with mean μ t and standard deviation σ t.
This algorithm can easily be adapted to compute the variance of a finite population: simply divide by n instead of n − 1 on the last line.. Because SumSq and (Sum×Sum)/n can be very similar numbers, cancellation can lead to the precision of the result to be much less than the inherent precision of the floating-point arithmetic used to perform the computation.
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If X = X * then the random variable X is called "real". An expectation E on an algebra A of random variables is a normalized, positive linear functional. What this means is that E[k] = k where k is a constant; E[X * X] ≥ 0 for all random variables X; E[X + Y] = E[X] + E[Y] for all random variables X and Y; and; E[kX] = kE[X] if k is a constant.
Poison Profits. A HuffPost / WNYC investigation into lead contamination in New York City