Download presentation

Presentation is loading. Please wait.

Published byHazel Lutman Modified over 3 years ago

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

Similar presentations

OK

Probability Theory Overview and Analysis of Randomized Algorithms Prepared by John Reif, Ph.D. Analysis of Algorithms.

Probability Theory Overview and Analysis of Randomized Algorithms Prepared by John Reif, Ph.D. Analysis of Algorithms.

© 2018 SlidePlayer.com Inc.

All rights reserved.

To ensure the functioning of the site, we use **cookies**. We share information about your activities on the site with our partners and Google partners: social networks and companies engaged in advertising and web analytics. For more information, see the Privacy Policy and Google Privacy & Terms.
Your consent to our cookies if you continue to use this website.

Ads by Google

Probability for kids ppt on batteries Ppt on waxes definition Ppt on history of google Ppt on indian textile industries in pakistan Download ppt on coordinate geometry for class 9th biology Ppt on production function Ppt on organ donation in india Ppt on water conservation in hindi Ppt on zener diode circuit Ppt on uses of concave and convex mirror