1. Let X 1, X 2,..., X n be a set of independent random variables having a common distribution, and let E[ X i ] = . then, with probability 1 Strong law of large numbers

2. Let X 1, X 2,..., X n be a set of independent random variables having a common distribution with mean  and variance  Then the distribution of Central Limit Theorem

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

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

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

8. 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 X i that each customer i ( i = 1, 2,..., N ) purchases is also a random variable E [ X i ] with expected value E [ X i ].

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 X i that each customer i ( i = 1, 2,..., N ) purchases is also a random variable E [ X i ] with expected value E [ X i ]. 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 X i that each customer i ( i = 1, 2,..., N ) purchases is also a random variable E [ X i ] with expected value E [ X i ]. 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 [ X i ]= E [ X ] and variance Var[ X i ]= 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

25. If N is Poisson distributed with parameter, the random Y = X 1 + X 2 +...+ X N is called a compound Poisson random variable

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

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

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

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

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

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