3.4.1 Covariance Flashcards
(20 cards)
What does covariance measure?
How two random variables change together — whether increases in one tend to correspond with increases or decreases in the other.
What is the formula for covariance?
Cov(X, Y) = E[(X - E[X]) * (Y - E[Y])]
What is the shortcut formula for covariance?
Cov(X, Y) = E[XY] - E[X] * E[Y]
What does positive covariance mean?
When X increases, Y tends to increase too.
What does negative covariance mean?
When X increases, Y tends to decrease.
Can covariance be negative?
Yes — unlike variance, which is always non-negative.
What is Cov(X, X)?
Var(X) — the covariance of a variable with itself is its variance.
What is Cov(c, X) or Cov(c₁, c₂)? (where c is a constant)
0 — constants have no variability.
What happens when you factor a constant out of a covariance?
Cov(aX, bY) = ab * Cov(X, Y)
Do additive constants affect covariance?
No — Cov(X + c, Y) = Cov(X, Y)
Is covariance distributive?
Yes — Cov(X + Y, Z) = Cov(X, Z) + Cov(Y, Z)
What is the formula for Var(X + Y) when X and Y are not independent?
Var(X + Y) = Var(X) + Var(Y) + 2 * Cov(X, Y)
How do you compute covariance from a joint distribution?
Use: Cov(X, Y) = E[XY] - E[X] * E[Y], where each expectation is computed from the joint PMF or PDF.
Why might E[XY] be easier to compute than Cov(X, Y) directly?
Because E[XY] uses raw moments, not central moments — easier to compute from tables.
How many covariance terms do you need for 3 variables X, Y, Z?
3 terms: Cov(X,Y), Cov(X,Z), Cov(Y,Z)
What happens to Cov(X, Y) if X and Y are independent?
Cov(X, Y) = 0
Does Cov(X, Y) = 0 imply X and Y are independent?
No — zero covariance implies uncorrelated, but not necessarily independent.
What should you never assume on an exam?
Independence — unless explicitly stated or proven.
If Cov(X, Y) = 0 but X and Y are not independent, what does that mean?
There is no linear relationship, but they might still be dependent in a nonlinear way.
How can you show independence to prove Cov(X, Y) = 0?
Show f(x, y) = f_X(x) * f_Y(y) and that the ranges of X and Y do not restrict each other.
