Random Variables (15) Flashcards
This assumes any of several different numeric values as a result of some random event. These are denoted by a capital letter such as X.
Random variable
This is a random variable that can take on of a finite (or countable) number of distinct outcomes.
Discrete random variable
This is a random variable that can take any numeric value within a range of values. The range may be infinite or bounded at either or both ends.
Continuous random variable
This is a function that associates a probability P with each value of a discrete random variable X, denoted P(X=x), or with any interval of values of a continuous random variable.
Probability model
This is a random variables theoretical long-run average value, the center of its model. Denoted µ or E(X), it is found (if the random variable is discrete) by summing the products of variable values and probabilities.
Definition
Expected Value
Expected Value
Formula
µ = E(X) = ΣxP(x)
Sum of the products of variable values and probabilities.
This is the expected value of the squared deviation from the mean.
Definition
Variance
σ^2
Variance
Formula for σ^2
σ^2 = Var(X) = Σ(x-µ)^2 P(x)
This describes the spread in the model and is the square root of the variance.
σ = SD(X) = √ Var(X)
Changing a random variable by a constant
Four Probability Formulas
E(X ± c) = E(x) ± c
Var(X ± c) = Var(X)
E(aX) = aE(X)
Var(aX) = a^2Var(x)
Adding of Subtracting random variables
Two Formulas
E(X ± Y) = E(X) ± c
Var(X ± c) = Var(X)
E(aX) = aE(X)
Var(aX) = a^2Var(X)
If X and Y are independent
The Pythagorean Theorem of Statistics
Covariance
Formula
Cov(X,Y) = E((X - µ)(Y - v))
where µ=E(x) and v=E(Y)
Sum/Difference of Variances
Formula
Var(X±Y) = Var(X) + Var(Y) ± 2Cov(X,Y)
General Formula (no need to assume independence)
