1. Probability Theory
STAT 312
STAT 312
Dr. Zakeia AlSaiary

2. Theorem: 7.1. p 191
A real valued function f of two variables is joint probability density function of a pair of discrete random variables X and Y if and only if :

3. Example:7.1 page 191
For what value of the constant k the function given by Is a joint probability density function of some random variables X , Y ?

4. Marginal probability density function Example:

5. 7.2. Bivariate Continuous Random Variables
الإثنين 10/4/1435 المحاضرة الثالثة:
7.2. Bivariate Continuous
Random Variables

6. 7.2. Bivariate Continuous Random Variables
In this section, we shall extend the idea of probability density functions
of one random variable to that of two random variables.
Definition 7.5. The joint probability density function of the random variables X and Y is an integrable function f(x, y) such that

7. 7.2. Bivariate Continuous Random Variables
Example 7.6. Let the joint density function of X and Y be given by
What is the value of the constant k ?

9. REMARK:
If we know the joint probability density function f of the random variables X and Y , then we can compute the probability of the event A from:

10. Bivariate Continuous Random Variables
Example 7.7. Let the joint density of the ontinuous
random variables X and Y be
What is the probability of the event

12. Marginal probability density function:
Definition 7.6. Let (X, Y ) be a continuous bivariate random variable. Let f(x, y) be the joint probability density function of X and Y . The function
is called the marginal probability density function of X. Similarly, the function

13. Marginal probability density function:
Similarly, the function
is called the marginal probability density function of Y.

14. Marginal probability density function:
Example 7.9. If the joint density function for X and Y is given by:
What is the marginal density of X where nonzero?

16. Definition 7.7. Let X and Y be the continuous random variables with joint probability density function f(x, y). The joint cumulative distribution function F(x, y) of X and Y is defined as

17. The joint cumulative distribution function F(x, y):
From the fundamental theorem of calculus, we again: obtain

18. The joint cumulative distribution function F(x, y):
Example If the joint cumulative distribution function of X and Y is given by
then what is the joint density of X and Y ?

19. EXERCISES:
Page , 2 , 3 , 4 , 7 , 8 , 10 , 11

20. 7.3. Conditional Distributions
First, we motivate the definition of conditional distribution using discrete random variables and then based on this motivation we give a general definition of the conditional distribution. Let X and Y be two discrete random variables with joint probability density f(x, y).

21. 7.3. Conditional Distributions
Then by definition of the joint probability density, we have f(x, y) = P(X = x, Y = y). If A = {X = x}, B = {Y = y} and f (y) = P(Y = y), then from the above equation we have P ({X = x} / {Y = y}) = P (A/B)

22. 7.3. Conditional Distributions
If we write the P ({X = x} / {Y = y}) as g(x / y), then we have

23. 7.3. Conditional Distributions
Definition 7.8. Let X and Y be any two random variables with joint density f(x, y) and marginals f1(x) and f2(y). The conditional probability density function g of X, given (the event) Y = y, is defined as

24. 7.3. Conditional Distributions
Similarly, the conditional probability density function h of Y , given (the event) X = x, is defined as

25. 7.3. Conditional Distributions
Example Let X and Y be discrete random variables with joint probability function
What is the conditional probability density function of Y, given X = 2 ?

27. 7.3. Conditional Distributions
Example Let X and Y be discrete random variables with joint probability function
What is the conditional probability density function of Y, given X = x ?

29. 7.3. Conditional Distributions
Example Let X and Y be contiuous random variables with joint pdf
What is the conditional probability density function of Y, given X = x ?

31. 7.3. Conditional Distributions
Example Let X and Y random variables such that X has pdf
and the conditional density of Y given X = x is

32. 7.3. Conditional Distributions
What is the conditional density of X given Y = y over the appropriate domain?

33. 7.4. Independence of Random Variables
In this section, we define the concept of stochastic independence of two random variables X and Y . The conditional robability density function g of X given Y = y usually depends on y. If g is independent of y, then the random variables X and Y are said to be independent. This motivates the following definition.

34. 7.4. Independence of Random Variables
Definition 7.8. Let X and Y be any two random variables with joint density f(x, y) and marginals f1(x) and f2(y). The random variables X and Y are (stochastically) independent if and only if

35. 7.4. Independence of Random Variables
Example Let X and Y be discrete random variables with joint density
Are X and Y stochastically independent?

37. 7.4. Independence of Random Variables
Example Let X and Y have the joint density
Are X and Y stochastically independent?

39. 7.4. Independence of Random Variables
Example Let X and Y have the joint density
Are X and Y stochastically independent?

40. 7.4. Independence of Random Variables
Definition 7.9. The random variables X and Y are said to be independent and identically distributed (IID) if and only if they are independent and have the same distribution.

41. EXERCISES:
Page , 16 , 21