Distribution functions, Moments, Moment generating functions in the Multivariate case

The distribution function F(x) This is defined for any random variable, X. F(x) = P[X ≤ x] Properties F(-∞) = 0 and F(∞) = 1. F(x) is non-decreasing (i. e. if x1 < x2 then F(x1) ≤ F(x2) ) F(b) – F(a) = P[a < X ≤ b].

Discrete Random Variables F(x) p(x) F(x) is a non-decreasing step function with

Continuous Random Variables Variables F(x) f(x) slope x F(x) is a non-decreasing continuous function with To find the probability density function, f(x), one first finds F(x) then

The joint distribution function F(x1, x2, …, xk) is defined for k random variables, X1, X2, … , Xk. F(x1, x2, … , xk) = P[ X1 ≤ x1, X2 ≤ x2 , … , Xk ≤ xk ] for k = 2 x2 (x1, x2) x1 F(x1, x2) = P[ X1 ≤ x1, X2 ≤ x2]

Properties F(x1 , -∞) = F(-∞ , x2) = F(-∞ , -∞) = 0 F(x1 , ∞) = P[ X1 ≤ x1, X2 ≤ ∞] = P[ X1 ≤ x1] = F1 (x1) = the marginal cumulative distribution function of X1 F(∞, x2) = P[ X1 ≤ ∞, X2 ≤ x2] = P[ X2 ≤ x2] = F2 (x2) = the marginal cumulative distribution function of X2 F(∞, ∞) = P[ X1 ≤ ∞, X2 ≤ ∞] = 1

i.e. if a1 < b1 if a2 < b2 then F(a1, x2) ≤ F(b1 , x2) F(x1, x2 ) is non-decreasing in both the x1 direction and the x2 direction. i.e. if a1 < b1 if a2 < b2 then F(a1, x2) ≤ F(b1 , x2) F(x1, a2) ≤ F(x1 , b2) F( a1, a2) ≤ F(b1 , b2) x2 (b1, b2) (a1, b2) x1 (a1, a2) (b1, a2)

P[a < X1 ≤ b, c < X2 ≤ d] = F(b,d) – F(a,d) – F(b,c) + F(a,c). x2 (b, d) (a, d) x1 (a, c) (b, c)

Discrete Random Variables x2 (x1, x2) x1 F(x1, x2) is a step surface

Continuous Random Variables x2 (x1, x2) x1 F(x1, x2) is a surface

Non-central and Central Multivariate Moments Non-central and Central

Definition Let X1 and X2 be a jointly distirbuted random variables (discrete or continuous), then for any pair of positive integers (k1, k2) the joint moment of (X1, X2) of order (k1, k2) is defined to be:

Definition Let X1 and X2 be a jointly distirbuted random variables (discrete or continuous), then for any pair of positive integers (k1, k2) the joint central moment of (X1, X2) of order (k1, k2) is defined to be: where m1 = E [X1] and m2 = E [X2]

Note = the covariance of X1 and X2.

Multivariate Moment Generating functions

Recall The moment generating function

Definition Let X1, X2, … Xk be a jointly distributed random variables (discrete or continuous), then the joint moment generating function is defined to be:

Definition Let X1, X2, … Xk be a jointly distributed random variables (discrete or continuous), then the joint moment generating function is defined to be:

Power Series expansion the joint moment generating function (k = 2)

The Central Limit theorem revisited

The Central Limit theorem If x1, x2, …, xn is a sample from a distribution with mean m, and standard deviations s, then if n is large has a normal distribution with mean and variance

The Central Limit theorem illustrated If x1, x2 are independent from the uniform distirbution from 0 to 1. Find the distribution of: let

Now

Now: The density of

n = 1 1 n = 2 1 n = 3 1