Covariance(X,Y) formula
Cov(X, Y) = E[XY] - E[X].E[Y]
Correlation: ρ(X, Y) formula
Cov(X, Y) / sqrt(Var(X).Var(Y))
Covariance and correlation of independent variables
0
Var(aX + bY) formula
a^2 Var(X) + b^2 Var(Y) + 2ab.Cov(X, Y)
Var(aX-bY) formula
a^2 Var(X) + b^2 Var(Y) - 2ab.Cov(X, Y)
Covariance relation with correlation
ρ(X, Y) = Cov(X, Y) / (σ_X * σ_Y)
Linearity of Expectations
E[aX + bY]
E[aX + bY] = aE[X] + bE[Y]
Var(XY) formula
Var(XY) = E[X^2 Y^2] - (E[XY])^2
E[XY] for independent
E[X]. E[Y]
Martingale Property
A stochastic process {X_t} is a martingale w.r.t. a filtration {F_t} if:
- E[|X_t|] < ∞ for all t
- E[X_{t+1} | F_t] = X_t
Markov Property
A stochastic process {X_t} is Markov if:
P(X_{t+1} | X_t, X_{t-1}, …, X_0) = P(X_{t+1} | X_t)
Martingale Property
A stochastic process {X_t} is a martingale w.r.t. a filtration {F_t} if:
- E[|X_t|] < ∞ for all t
- E[X_{t+1} | F_t] = X_t
Symmetric random walk starting from 0 that stops at α or -β. P(α before β) formula
P(reaching α before -β) = β / (α + β),
where reaching α before -β,
β > 0,
α > 0
Cov(X, X)
Var(X)
Cov(cX, Y)
c.Cov(X, Y)
Cov(X, Y+Z)
Cov(X, Y) + Cov(X, Z)
Conditional expectation of r.v.
E[X | Y = y] = ∫{-∞}^{∞} x · f{X|Y}(x | y) dx
where
f_{X|Y}(x | y) = f_{X,Y}(x, y) / f_Y(y)
Inclusion Exclusion Principle. Eg: 3 couples sitting in linear line. Find P of 0 couples sitting next to each other
P = 0 sitting besides - 1 forced sitting + 2 forced sitting - 3 forced sitting.
Sol -> 6! - C(3,1).(2^1).(5!) + C(3, 2).(2^2).4! - C(3,3).(2^3).3!
Cdf of standard normal = f(x)
What is f(-x)?
1 - f(x)
Z score formula
X - U / sigma
Characteristic function formula
E[exp(itx)], where i^2=(-1)
Stars and bars approach. No. of non-negative solutions to x1 + x2 + … + xr = n?
C(n + r - 1, r - 1)
Eg: 10 candies, 4 children
Ans: C(10 + 4 - 1, 4 - 1) = C(13, 3)
