Discrete & Continuous Distributions Flashcards
General formula for E(X) for discrete distributions
E(X) = sum of X P(X)
For discrete distributions, the cumulative distribution function =
CDF = sum of PDF
For continuous distributions, the cumulative distribution function =
CDF = integral of PDF
E(X + a) =
E(X) + a
E(aX) =
a E(X)
V(X) formula
V(X) = E[(X - E(X))^2] = E(X^2) - [E(X)]^2
V(X + a) =
V(X) - adding a constant doesn’t affect variance
V(aX) =
a^2 V(X) since variance is a squared operator
E(X + Y) =
E(X) + E(Y)
V(X + Y) =
V(X) + V(Y) + 2COV(X, Y)
COV(X, Y) =
E(XY) - E(X)E(Y)
V(X - Y) =
V(X) + V(Y) - 2COV(X, Y)
V(aX - bY) =
a^2 V(X) + b^2 V(Y) - 2ab COV(X, Y)
What is the covariance if X & Y are independent?
COV(X, Y) = 0
COV(X + a, Y) =
COV(X, Y)
COV(aX, Y) =
aCOV(X, Y)
Conditional probabilities must sum to…
1
How to work out probability between an interval for continuous distributions
Integrate f(X) dX between the interval [a, b]
Variance for a continuous distribution =
Integral of X^2 f(X) dX - [integral of X f(X) dX]^2
What is the expectation of the sample mean when we have 3 drawings?
E(X1 + X2 + X3/3) = 1/3 [E(X1) + E(X2) + E(X3)] = 1/3 x 3M = M
What is the variance of a sample mean when we have 3 drawings?
V(X1 + X2 + X3/3) = 1/9 [V(X1) + V(X2) + V(X3)] = 3sigma^2/9 = sigma^2 / 3
Covariance zero as independent random samples
