Preliminary Course
Applied Mathematics
Probabilities. A few properties
A few properties
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Expected value and variance
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For any constant c, we have E[c] = c
The expectation operator is linear, i.e. for any random variables X and Y and any constant c, we have:
E[ c X + Y] = c E[X] + E[Y]
Concerning now the variance:
VAR[a X + b] = a2 VAR[X]
and
VAR[X + Y] = VAR[X] + VAR[Y] + 2 Cov(X,Y)
where Cov(X,Y) = E[XY] - E[X] E[Y] is the covariance of X and Y
Law of Large Numbers
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Let Xi, i=1, ..., n be independent and identically distributed random variables,
with finite expected absolute value m= E(|X1|).
Let Sn = X1 + ... + Xn.
Then, Sn/n tends toward m almost surely, i.e.:
It means that if the number of trials grows to infinity, then the empirical mean of the sample converges toward the expected value of the random variable.
This law can be put in relation with the classical method of Monte-Carlo simulation that uses repeated samplings to determine the properties of some phenomena.