Week 4: Expected Value Covariance Flashcards
If Random Variable X takes finitely many values x1,x2..xn. with equal (uniform probability), the average of X is:
(1/n)*(sum of x values), where n is the number of values
How to calculate weighted average when Random Variable X takes finitely many values x1,x2..xn. with equal (uniform probability).
Weighted average = sum(pi*xi) , p is probability and x a value (index of value in set)
What is the expected value of a continuous random variable?
Mean value of outcomes over an interval.
E(X) = Integral (range infinity to negative infinity) x fx(x) dx
- Where fx(x) is the probability density function (PDF) of X
What does it mean for a random variable to be integrable?
Integrable means random variable expected value formula (discrete or continuous) converges (integral has a finite answer).
What does 1(a, b)(x) = to?
X is 1 when between a and b, and 0 otherwise.
E[aX + b] =
aE[X] + b
Expected value of a product of independent random variables is:
The product of the expected values
Variance formula given X is real valued random variable and mean = E(X)
Var(X) = E[(X - mean)^2], if discrete
Var(X) = Integral (range infinity to negative infinity) (x - mean)^2 fx(x) dx, if continuous
Using Variance translation theorem, if X satisfies E(X) = 0 (X is centered), what is variance?
Var(X) = E(X^2), because E(X) = mean = 0
Variance Translation Theorem
Var(X) = E(X^2) - (Mean^2) = E(X^2) - E(X)^2
If X is a real-valued random variable, then Var(aX + b) =
(a^2) * Var(X)
Covariance Formula
Cov(X, Y) = E[(X - mean x)(Y - mean y)]
Correlation formula (between two random variables)
Px,y = Cov(X, Y) / (stdX * StdY)
Px,y = E[(X - mean x)(Y - mean y)] / (stdX * StdY)
What happens to correlation if Cov(X, Y)?
If Cov(X, Y) = 0, Pxy=0, which means correlation is 0. Therefore X and Y are uncorrelated.
What does independence mean for correlation?
Indepence = Uncorrelated (but remember uncorrelated does not mean independent)