Strong law of large numbers

1. Strong law of large numbers Let X1, X2, ..., Xn be a set of independent random variables having a common distribution, and let E[Xi] = m. then, with probability 1

2. Central Limit Theorem Let X1, X2, ..., Xn be a set of independent random variables having a common distribution with mean m and variance s. Then the distribution of

3. Conditional probability and conditional expectations Let X and Y be two discrete random variables, then the conditional probability mass function of X given that Y=y is defined asfor all values of y for which P(Y=y)>0.

4. Conditional probability and conditional expectations Let X and Y be two discrete random variables, then the conditional probability mass function of X given that Y=y is defined asfor all values of y for which P(Y=y)>0.The conditional expectation of X given that Y=y is defined as

5. Let X and Y be two continuous random variables, then the conditional probability density function of X given that Y=y is defined asfor all values of y for which fY(y)>0.

6. Let X and Y be two continuous random variables, then the conditional probability density function of X given that Y=y is defined asfor all values of y for which fY(y)>0.The conditional expectation of X given that Y=y is defined as

8. Proof

9. Proof

10. Proof

11. Proof

12. Proof

13. Proof

14. The sum of a random number of random variables Example: The number N of customers that place orders each day with an online bookstore is a random variable with expected value E[N].

15. The sum of a random number of random variables Example: The number N of customers that place orders each day with an online bookstore is a random variable with expected value E[N]. The number of books Xithat each customer i (i = 1, 2, ..., N) purchases is also a random variable E[Xi] with expected value E[Xi].

16. The sum of a random number of random variables Example: The number N of customers that place orders each day with an online bookstore is a random variable with expected value E[N]. The number of books Xithat each customer i (i = 1, 2, ..., N) purchases is also a random variable E[Xi] with expected value E[Xi].What is the expected value of the total number of books Y sold each day? What is its variance?

17. The sum of a random number of random variables Example: The number N of customers that place orders each day with an online bookstore is a random variable with expected value E[N]. The number of books Xithat each customer i (i = 1, 2, ..., N) purchases is also a random variable E[Xi] with expected value E[Xi].What is the expected value of the total number of books Y sold each day? What is its variance? Assume that the number of books are independent and identically distributed with the same mean E[Xi]=E[X] and variance Var[Xi]=E[X] for i=1,..., N. Also assume the number of books purchased per customer is independent of the total number of customers.

18. The expected value

19. The variance

20. The variance

21. The variance

22. The variance

23. The variance

25. If N is Poisson distributed with parameter l, the random Y = X1+X2+...+ XN is called a compound Poisson random variable

26. Computing probabilities by conditioning Let E denote some event. Define a random variable X by

27. Computing probabilities by conditioning Let E denote some event. Define a random variable X by

28. Computing probabilities by conditioning Let E denote some event. Define a random variable X by

29. Example 1: Let X and Y be two independent continuous random variables with densities fX and fY. What is P(X<Y)?

30. Example 1: Let X and Y be two independent continuous random variables with densities fX and fY. What is P(X<Y)?

31. Example 1: Let X and Y be two independent continuous random variables with densities fX and fY. What is P(X<Y)?

32. Example 1: Let X and Y be two independent continuous random variables with densities fX and fY. What is P(X<Y)?

33. Example 2: Let X and Y be two independent continuous random variables with densities fX and fY. What is the distribution of X+Y?

34. Example 2: Let X and Y be two independent continuous random variables with densities fX and fY. What is the distribution of X+Y?

35. Example 2: Let X and Y be two independent continuous random variables with densities fX and fY. What is the distribution of X+Y?

36. Example 2: Let X and Y be two independent continuous random variables with densities fX and fY. What is the distribution of X+Y?

37. Example 2: Let X and Y be two independent continuous random variables with densities fX and fY. What is the distribution of X+Y?

38. Example 2: Let X and Y be two independent continuous random variables with densities fX and fY. What is the distribution of X+Y?

39. Example 2: Let X and Y be two independent continuous random variables with densities fX and fY. What is the distribution of X+Y?

40. Example 3: (Thinning of a Poisson) Suppose X is a