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

2. 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].

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

4. 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

5. 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]

6. 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

7. 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)

8. 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)

9. Discrete Random Variablesx2
(x1, x2) x1
F(x1, x2) is a step surface

10. Continuous Random Variablesx2
(x1, x2) x1
F(x1, x2) is a surface

11. Non-central and Central
Multivariate Moments
Non-central and Central

12. 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:

13. 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]

14. Note
= the covariance of X1 and X2.

15. Multivariate Moment Generating functions

16. Recall
The moment generating function

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

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

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

20. The Central Limit theorem
revisited

21. 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

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

23. Now

24. Now:
The density of

25. n = 1 1
n = 2 1
n = 3 1