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Saturday, February 17, 2024

Spaces in Math


Probability Space

Probability Space = Sample Space + Events + Probabilities of events

Metric Space


Vector Space

Eculdien Space

Hilbert Space


Distance Measures



X = (1,6,2,5,3)

Y = (3,5,2,6,6)

Euclidian distance

distance = 

distance = 3.87

Manhattan distance

distance = 
distance = 7

Chebychev distance

distance = 
distance  = 3

Expectation vs average


Let  represent the outcome of a roll of an unbiased six-sided die. The possible values for  are 1, 2, 3, 4, 5, and 6, each having the probability of occurrence of 1/6. The expectation value (or expected value) of  is then given by

()expected=1(1/6)+2(1/6)+3(1/6)+4(1/6)+5(1/6)+6(1/6)=21/6=3.5

Suppose that in a sequence of ten rolls of the die, if the outcomes are 5, 2, 6, 2, 2, 1, 2, 3, 6, 1, then the average (arithmetic mean) of the results is given by

()average=(5+2+6+2+2+1+2+3+6+1)/10=3.0

We say that the average value is 3.0, with the distance of 0.5 from the expectation value of 3.5. If we roll the die  times, where  is very large, then the average will converge to the expected value, i.e.,()average=()expected. This is evidently because, when  is very large each possible value of  (i.e. 1 to 6) will occur with equal probability of 1/6, turning the average to the expectation value.


Resource: stackoverflow