Distribution functions Moment generating functions in the Multivariate

Distribution functions, 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 1. 2. 3. F(-∞) = 0 and F(∞) = 1. F(x) is non-decreasing (i. e. if x 1 < x 2 then F(x 1) ≤ F(x 2) ) F(b) – F(a) = P[a < X ≤ b].

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

5. Continuous Random Variables f(x) slope F(x) 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(x 1, x 2, …, xk) is defined for k random variables, X 1, X 2, … , Xk. F(x 1, x 2, … , xk) = P[ X 1 ≤ x 1, X 2 ≤ x 2 , … , Xk ≤ xk ] for k = 2 x 2 (x 1, x 2) x 1 F(x 1, x 2) = P[ X 1 ≤ x 1, X 2 ≤ x 2]

Properties 1. 2. F(x 1 , -∞) = F(-∞ , x 2) = F(-∞ , -∞) = 0 F(x 1 , ∞) = P[ X 1 ≤ x 1, X 2 ≤ ∞] = P[ X 1 ≤ x 1] = F 1 (x 1) = the marginal cumulative distribution function of X 1 F(∞, x 2) = P[ X 1 ≤ ∞, X 2 ≤ x 2] = P[ X 2 ≤ x 2] = F 2 (x 2) = the marginal cumulative distribution function of X 2 F(∞, ∞) = P[ X 1 ≤ ∞, X 2 ≤ ∞] = 1

3. F(x 1, x 2 ) is non-decreasing in both the x 1 direction and the x 2 direction. i. e. if a 1 < b 1 if a 2 < b 2 then i. F(a 1, x 2) ≤ F(b 1 , x 2) ii. F(x 1, a 2) ≤ F(x 1 , b 2) iii. F( a 1, a 2) ≤ F(b 1 , b 2) x 2 (a 1, b 2) (b 1, b 2) x 1 (a 1, a 2) (b 1, a 2)

4. Discrete Random Variables x 2 (x 1, x 2) x 1 F(x 1, x 2) is a step surface

5. Continuous Random Variables x 2 (x 1, x 2) x 1 F(x 1, x 2) is a surface

Multivariate Moments Non-central and Central

Definition Let X 1 and X 2 be a jointly distirbuted random variables (discrete or continuous), then for any pair of positive integers (k 1, k 2) the joint moment of (X 1, X 2) of order (k 1, k 2) is defined to be:

Definition Let X 1 and X 2 be a jointly distirbuted random variables (discrete or continuous), then for any pair of positive integers (k 1, k 2) the joint central moment of (X 1, X 2) of order (k 1, k 2) is defined to be: where m 1 = E [X 1] and m 2 = E [X 2]

Note = the covariance of X 1 and X 2.

Multivariate Moment Generating functions

Recall The moment generating function

Definition Let X 1, X 2, … Xk be a jointly distributed random variables (discrete or continuous), then the joint moment generating function is defined to be:

Definition Let X 1, X 2, … 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 x 1, x 2, …, 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 x 1, x 2 are independent from the uniform distirbution from 0 to 1. Find the distribution of: let

Now

Now: The density of

n=1 0 1 n=2 0 1 n=3 0 1