Expectation

E[X] = ΣxP(X = x) = ∫xp(x)dx  [p() is pdf here ]

Clearly E[f(X)] = Σ y P(Y = y ) [y = f(x)] = Σ f(x) P(X = x) = ∫f(x)p(x)dx [By creating mapping from Y to X]

As P(Y = y) = Σ P(X = x )
                [∀ x s.t f(x) = y]

If X,Y mutually exclusive  E[X+Y] = E[X] + E[Y] [from above]

Some candy expectations :-
V[X] (Variance) = E[X^2 - E[X]] = σ(X)^2 [standard deviation]
E[X] = μ(X) [mean]

Moreover
x s.t P(X< x) = 1/2 [median]
x s.t P(X = x) = max P(X = s) [mode]
                           ∀s

E[X^n] =  n'th moment of X.

Moment Generating Function:
M(X,t) = E[e^(xt)] is the moment generating function where ∂^n(M(x,t))/(∂t)^n  | t = 0 =  n'th moment

Probability Generating Function :

G(X,z) = E[z^x] and 1/n! (∂^n(M(x,z))/(∂z)^n) | z = 0 is the P[X = n]

Covariance : Cov(X,Y) = E[(X - E[X])(Y-E[Y])] = E[XY] - E[X]E[Y]

Lemma :
V(X+Y) = V(X) + V(Y) - Cov(X,Y) = E[(X+Y)^2] - E[(X+Y)]^2

Cov(X,Y) = 0 => E[XY] = E[X]E[Y]  <=> X ⊥ Y