Module 5: Multivariate Normal Distribution

A variable X follows a discrete probability distribution if the possible values of X are either:

• A finite set
• A countable infinite sequence

p_x(x_i) = P(X=x_i) is called the probability mass function (PMF)

• p_x(x_i)  >= 0 as it is a probability
  
• The sum of PMF for all values of X = 1

Moment Generating Function

Moments, such as E(X), V(X), can also be calculated using the Moment Generating Function (MGF):

The rth moment of X, E(X^r) can be obtained by differentiating M_x(t) r times with respect to t and setting t=0

• M_x(0) = 1
• M^I_x(0) = E(X)
• M^II_x(0) = E(X²) -> V(X) = M^II_x(0) - (MI_x(0))2
• In general, M_x^(r)(0) = E(X^r)

Uniqueness: if X and Y are two random variables and M_x(t) = M_y(t) when |t| < h for some positive number h, then X and Y have the same distribution

Note: MGF does not exist for all distributions (E(e^tx) may be infinity)