Download presentation

Presentation is loading. Please wait.

Published byClarence Small Modified over 2 years ago

1
Introduction to Probability and Statistics Chapter 5 Discrete Distributions

2
Discrete Random Variables Discrete random variables take on only a finite or countable many of values. Number of heads in 1000 trials of coin tossing Number of cars that enter UNI in a certain day Number of heads in 1000 trials of coin tossing Number of cars that enter UNI in a certain day

3
Binomial Random Variable coin-tossing experiment binomial random variable.The coin-tossing experiment is a simple example of a binomial random variable. Toss a fair coin n = 3 times and record x = number of heads. xp(x)p(x) 01/8 13/8 2 31/8

4
Example Toss a coin 10 times For each single trial, probability of getting a head is 0.4 Let x denote the number of heads

5
The Binomial Experiment n identical trials. 1.The experiment consists of n identical trials. one of two outcomes 2.Each trial results in one of two outcomes, success (S) or failure (F). remains constant 3.Probability of success on a single trial is p and remains constant from trial to trial. The probability of failure is q = 1 – p. independent 4.Trials are independent. x, the number of successes in n trials. 5.Random variable x, the number of successes in n trials. x – Binomial random variable with parameters n and p x – Binomial random variable with parameters n and p

6
Binomial or Not? A box contains 4 green M&Ms and 5 red ones Take out 3 with replacement x denotes number of greens Is x binomial? Yes, 3 trials are independent with same probability of getting a green. m m mm mm

7
Binomial or Not? A box contains 4 green M&Ms and 5 red ones Take out 3 without replacement x denotes number of greens Is x binomial? NO, when we take out the second M&M, the probability of getting a green depends on color of the first. 3 trials are dependent. m m mm mm

8
Binomial or Not? Very few real life applications satisfy these requirements exactly. Select 10 people from the U.S. population, and suppose that 15% of the population has the Alzheimer’s gene. For the first person, p = P(gene) =.15 For the second person, p P(gene) =.15, even though one person has been removed from the population… For the tenth person, p P(gene) =.15 Yes, independent trials with the same probability of success

9
Binomial Random Variable Example:Example: A geneticist samples 10 people and x counts the number who have a gene linked to Alzheimer’s disease. Success:Success: Failure:Failure: Number ofNumber of trials: trials: Probability of SuccessProbability of Success Has gene Doesn’t have gene n = 10 p = P(has gene) = 0.15 Rule of Thumb: Sample size n; Population size N; If n/N <.05, the experiment is Binomial.

10
Example Toss a coin 10 times For each single trial, probability of getting a head is 0.4 Let x denote the number of heads Find probability of getting exactly 3 heads. i.e. P(x=3). Find probability distribution of x

11
Solution Simple events: Event A: {strings with exactly 3 H’s}; Probability of getting a given string in A: Probability of event A. i.e. P(x=3) Number of strings in A Strings of H’s and T’s with length 10 HTTTHTHTTT TTHHTTTTHT… HTTTHTHTTT

12
A General Example Toss a coin n times; For each single trial, probability of getting a head is p; Let x denote the number of heads; Find the probability of getting exactly k heads. i.e. P(x=k) Find probability distribution of x.

13
Binomial Probability Distribution For a binomial experiment with n trials and probability p of success on a given trial, the probability of k successes in n trials is

14
Binomial Mean, Variance and Standard Deviation For a binomial experiment with n trials and probability p of success on a given trial, the measures of center and spread are:

15
n = p =x = success =Example A marksman hits a target 80% of the time. He fires 5 shots at the target. What is the probability that exactly 3 shots hit the target? 5.8hit# of hits

16
Example What is the probability that more than 3 shots hit the target?

17
Example x = number of hits. What are the mean and standard deviation for x? (n=5,p=.8)

18
Cumulative Probability cumulative probability tables You can use the cumulative probability tables to find probabilities for selected binomial distributions. Binomial cumulative probability: P(x k) = P(x = 0) +…+ P(x = k) Binomial cumulative probability: P(x k) = P(x = 0) +…+ P(x = k)

19
Key Concepts I. The Binomial Random Variable 1. Five characteristics: the experiment consists of n identical trials; each resulting in either success S or failure F; probability of success is p and remains constant; all trials are independent; x is the number of successes in n trials. 2. Calculating binomial probabilities a. Formula: b. Cumulative binomial probability P(x k). 3. Mean of the binomial random variable: 4. Variance and standard deviation:

20
Example According to the Humane Society of the United States, there are approximately 40% of U.S. households own dogs. Suppose 15 households are selected at random. Find 1.probability that exactly 8 households own dogs? 2.probability that at most 3 households own dogs? 3.probability that more than 10 own dogs? 4.the mean, variance and standard deviation of x, the number of households that own dogs.

21
n = p =x = success =Example According to the Humane Society of the United States, there are approximately 40% of U.S. households own dogs. Suppose 15 households are selected at random. What is probability that exactly 8 households own dogs? 15.4own dog# households that own dog

22
Example What is the probability that at most 3 households own dogs?

23
Example What are the mean, variance and standard deviation of random variable x? (n=15, p=.4)

24
Binomial Probability Probability distribution for Binomial random variable x with n=15, p=0.4

25
Example 1.What are the mean, variance and standard deviation of random variable x? 2.Calculate interval within 2 standard deviations of mean. What values fall into this interval? 3.Find the probability that x fall into this interval.

Similar presentations

OK

Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin Chapter 5 Discrete Random Variables.

Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin Chapter 5 Discrete Random Variables.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on economy of india Ppt on job evaluation and job rotation Ppt on viruses and bacteria images Ppt on stock market basics Ppt on curiosity rover Ppt on introduction to object-oriented programming in java Download ppt on the nationalist movement in indochina Ppt on social contract theory john Ppt on natural resources for class 5 Ppt on db2 mainframes ibm