C4: DISCRETE RANDOM VARIABLES

Slides:



Advertisements
Similar presentations
Random Variables ECE460 Spring, 2012.
Advertisements

Random Variable A random variable X is a function that assign a real number, X(ζ), to each outcome ζ in the sample space of a random experiment. Domain.
Discrete Probability Distributions
CIS 2033 based on Dekking et al. A Modern Introduction to Probability and Statistics Slides by Michael Maurizi Instructor Longin Jan Latecki C9:
Review of Basic Probability and Statistics
CSE 221: Probabilistic Analysis of Computer Systems Topics covered: Discrete random variables Probability mass function Distribution function (Secs )
Probability Distributions Finite Random Variables.
1 Review of Probability Theory [Source: Stanford University]
Probability Mass Function Expectation 郭俊利 2009/03/16
Probability Distributions Random Variables: Finite and Continuous Distribution Functions Expected value April 3 – 10, 2003.
CSE 221: Probabilistic Analysis of Computer Systems Topics covered: Special discrete distributions Sec
C4: DISCRETE RANDOM VARIABLES CIS 2033 based on Dekking et al. A Modern Introduction to Probability and Statistics Longin Jan Latecki.
Discrete Random Variables and Probability Distributions
Chapter 21 Random Variables Discrete: Bernoulli, Binomial, Geometric, Poisson Continuous: Uniform, Exponential, Gamma, Normal Expectation & Variance, Joint.
Probability Distributions: Finite Random Variables.
CIS 2033 based on Dekking et al. A Modern Introduction to Probability and Statistics, 2007 Instructor Longin Jan Latecki Chapter 7: Expectation and variance.
Binomial & Geometric Random Variables §6-3. Goals: Binomial settings and binomial random variables Binomial probabilities Mean and standard deviation.
Statistics for Engineer Week II and Week III: Random Variables and Probability Distribution.
Binomial Distributions Calculating the Probability of Success.
Random Variables. A random variable X is a real valued function defined on the sample space, X : S  R. The set { s  S : X ( s )  [ a, b ] is an event}.
CIS 2033 based on Dekking et al. A Modern Introduction to Probability and Statistics Michael Baron. Probability and Statistics for Computer Scientists,
Discrete Probability Distributions. Random Variable Random variable is a variable whose value is subject to variations due to chance. A random variable.
Binomial Experiment A binomial experiment (also known as a Bernoulli trial) is a statistical experiment that has the following properties:
Binomial Probability Distribution
COMP 170 L2 L17: Random Variables and Expectation Page 1.
King Saud University Women Students
Chapter Four Random Variables and Their Probability Distributions
Week 21 Conditional Probability Idea – have performed a chance experiment but don’t know the outcome (ω), but have some partial information (event A) about.
Math b (Discrete) Random Variables, Binomial Distribution.
STA347 - week 31 Random Variables Example: We roll a fair die 6 times. Suppose we are interested in the number of 5’s in the 6 rolls. Let X = number of.
Lesson 6 - R Discrete Probability Distributions Review.
Exam 2: Rules Section 2.1 Bring a cheat sheet. One page 2 sides. Bring a calculator. Bring your book to use the tables in the back.
Random Variables Learn how to characterize the pattern of the distribution of values that a random variable may have, and how to use the pattern to find.
Random Variables Example:
Probability Distributions, Discrete Random Variables
AP STATISTICS Section 7.1 Random Variables. Objective: To be able to recognize discrete and continuous random variables and calculate probabilities using.
C4: DISCRETE RANDOM VARIABLES CIS 2033 based on Dekking et al. A Modern Introduction to Probability and Statistics Longin Jan Latecki.
Engineering Probability and Statistics - SE-205 -Chap 3 By S. O. Duffuaa.
Random Variables Lecture Lecturer : FATEN AL-HUSSAIN.
CIS 2033 based on Dekking et al. A Modern Introduction to Probability and Statistics B: Michael Baron. Probability and Statistics for Computer Scientists,
Discrete and Continuous Random Variables Section 7.1.
Probability Distributions
Negative Binomial Experiment
Random variables (r.v.) Random variable
Random Variables.
Engineering Probability and Statistics - SE-205 -Chap 3
Chapter Four Random Variables and Their Probability Distributions
Samples and Populations
C14: The central limit theorem
The Binomial and Geometric Distributions
Chapter 10: Covariance and Correlation
Distributions and expected value
MATH 3033 based on Dekking et al
CIS 2033 based on Dekking et al
Chapter 3 : Random Variables
M248: Analyzing data Block A UNIT A3 Modeling Variation.
6: Binomial Probability Distributions
Lesson #5: Probability Distributions
Discrete Random Variables: Basics
Discrete Random Variables: Basics
Experiments, Outcomes, Events and Random Variables: A Revisit
C19: Unbiased Estimators
Chapter 10: Covariance and Correlation
MATH 3033 based on Dekking et al
1/2555 สมศักดิ์ ศิวดำรงพงศ์
MATH 3033 based on Dekking et al
Discrete Random Variables: Basics
Chapter 10: Covariance and Correlation
MATH 2311 Section 3.3.
Presentation transcript:

C4: DISCRETE RANDOM VARIABLES CIS 2033 based on Dekking et al. A Modern Introduction to Probability and Statistics. 2007 Slides by Emily LeBlanc Instructor Longin Jan Latecki

Discrete Random Variables Discrete random variables are obtained by counting and have sample spaces which Are countable. The values that represent each outcome are usually integers. Random variables are denoted by capital letters. X: is the number of times that we flip a coin until H comes up The possible outcomes are denoted by lower case letters: a=1, a=2, a=3...

Probability Mass Function The probability mass function of a discrete random variable maps each possible outcome in the sample space to it's corresponding probability. The sum of the probabilities of all possible outcomes will always be equal to 1.

Probability Distribution Function The distribution function of a random variable X (also referred to as the cumulative distribution function) gives us information regarding the probability that X will take a value less than or equal to a. The value of F(a) is equal to the sum of all probabilities of outcomes less than or equal to a.

Graphs of PMF and CDF Probability Distribution Function Cumulative Distribution Function

Bernoulli Distribution The Bernoulli distribution is used to model an experiment with only two outcomes, success and failure. The parameter p is the chance for success. An example is flipping a coin, where “heads” may be success and “tails” may be failure.

Binomial Distribution The Binomial Distribution represents multiple Bernoulli trials. The parameter n is the number of trials, and the parameter p is the probability of success as in the Bernoulli distribution. P(X=k) is the probability of k successful outcomes in n trials.

Geometric Distribution A geometric distribution gives information about the probability of success after k attempts. The parameter p is the probability of success on the kth try. This means that all previous k-1 tries failed An example of this would be finding the probability that you will hit a bullseye with a dart on your kth toss.