Karteikarten
IX. Random Variables
Karten in diesem Stapel (16)
- 1Random variableA functions that is defined on a sample space
- 2The probability distribution of the random variableThe function
P(X=xj) = f(xj)
is called the probability distribution of the random variable. - 3The joint probability distribution of two random variables defined on the same sample spaceThe function
P(X=xj, Y=yk) = p(xj,yk)
is called the joint probability distribution of X and Y. - 4Relation between the joint probability distribution and the individual probability distributionsp(xj, y1) + p(xj, y2) + ... = P(X=xj)
p(x1, yk) + p(x2, yk) + ... = P(Y=yk) - 5Stochastic independence of two random variablesX and Y are called stochastically independent if:
P(xj,yk) = P(xj)*P(yk) - 6The expected value (mean) of XE(X) = Sum [xk*f(xk)]
provided that the series converges absolutely. - 7Expectation of a function on XLet g(X) be a function. Than g(X) defines a new random variable G. if it has a finite expectation, then:
E(g(X)) = Sum [g(xk)*f(xk)] - 8Expectation of X1 + X2 + ... + XnFor any random variables X1, X2, ...,Xn:
E(X1 + X2 + ... + Xn) = E(X1) + E(X2) + ... + E(Xn) - 9Expectation of X*YIf X and Y are mutually independent random variables, then:
E(X*Y) = E(X)*E(Y) - 10The r-th moment of X about the origin. Existence of (r-1)th moment.E(X^r) = Sum [xj^r * f(xj)]
is called the r-th moment of X. If the series does not converge absolutely, we say that the r-th moment does not exist.
Whenever r-th moment exists, so do all the preceding moments. - 11The varience and the standard deviation of XLet X be a random variable with second moment E(X^2). Then
var(X) = E((X - EX)^2) = E(X^2) - E(X)^2
std(X) = sqrt(var(X)) - 12Variable normalizationX* = (X - E(X))/std(X)
- 13Covarience of X and Y. Special case if X and Y are independentcov(X,Y) = E((X-EX)*(Y-EY)) = E(X*Y) - E(X)*E(Y)
If X and Y are independent, then cov(X,Y)=0 - 14var(X1 + X2 + ... + Xn)var(X1+...+Xn) = Sum[var(Xj)] + 2*Sum[cov(Xj,Xk) for all pairs Xj, Xk]
- 15Chebyshev's inequalityFor any t > 0:
P{|X|>= t} <= t^-2 * E(X^2)
or:
P{|X-EX|>=t} <= t^-2 * var(X) - 16The correlation coefficientr = cov(X,Y)/(std(X)*std(Y))
-1 <= r <= 1
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- Probability Theory and It's Applications - W. Feller Karteikarten am IX. Random Variables, erstellt von Sergei Fomin am 22/08/2017.
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- Sergei Fomin
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