Chapter 6:
This presentation is the property of its rightful owner.
Sponsored Links
1 / 20

Chapter 6: Binomial Probability Distributions PowerPoint PPT Presentation


  • 75 Views
  • Uploaded on
  • Presentation posted in: General

Chapter 6: Binomial Probability Distributions. In Chapter 6:. 6.1 Binomial Random Variables 6.2 Calculating Binomial Probabilities 6.3 Cumulative Probabilities 6.4 Probability Calculators 6.5 Expected Value and Variance .. 6.6 Using the Binomial Distribution to Help Make Judgments.

Download Presentation

Chapter 6: Binomial Probability Distributions

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


October 12

Chapter 6: Binomial Probability Distributions


In chapter 6

In Chapter 6:

6.1 Binomial Random Variables

6.2 Calculating Binomial Probabilities

6.3 Cumulative Probabilities

6.4 Probability Calculators

6.5 Expected Value and Variance ..

6.6 Using the Binomial Distribution to Help Make Judgments


Binomial random variables

Binomial Random Variables

  • Bernoulli trial≡ a random event with two possible outcomes (“success” or “failure”)

  • Binomial random variable ≡ the random number of successes in n independent Bernoulli trials, each trial with the same probability of success

  • Binomials have two parameters:n  number of trialsp  probability of success of each trial


Binomials cont

Binomials (cont.)

  • Only two outcomes are possible (success and failure)

  • The outcome of each trial does not depend on the previous trial (independence)

  • The probability for success p is the same for each trial

  • Trials are repeated a specified number of times n


Calculating binomial probabilities by hand

Calculating Binomial Probabilities by hand

Formula:

where

nCx≡ the binomial coefficient (next slide)

p≡ probability of success for a single trial

q≡ probability of failure for single trial = 1 – p


Binomial coefficient

Binomial Coefficient

Formula for the binomial coefficient:

where ! represents the factorial function:

x!= x  (x – 1)  (x – 2)  …  1

For example, 4! = 4  3  2  1 = 24

By definition 1! = 1 and 0! = 1

For example:


Binomial coefficient1

Binomial Coefficient

The binomial coefficient tells you the number of ways you could choose x items out of n

nCx the number of ways to x items out of n

For example, 4C2 = 6 Therefore, there are 6 ways to choose 2 items out of 4.


Binomial calculation example

Binomial Calculation – Example

“Four patients example”: X ~ b(4,.75).

Note q = 1 −.75 = .25.

What is the probability of 0 successes?


X b 4 0 75 continued

X~b(4,0.75), continued

Pr(X = 1) = 4C1· 0.751 · 0.254–1

= 4 · 0.75 · 0.0156

= 0.0469

Pr(X = 2) = 4C2· 0.752 · 0.254–2

= 6 · 0.5625 · 0.0625

= 0.2106


X b 4 0 75 continued1

X~b(4, 0.75) continued

Pr(X = 3) = 4C3· 0.753 · 0.254–3

= 4 · 0.4219 · 0.25

= 0.4219

Pr(X = 4) = 4C4· 0.754 · 0.254–4

= 1 · 0.3164 · 1

= 0.3164


Pmf for x b 4 0 75 tabular and graphical forms

pmf for X~b(4, 0.75)Tabular and graphical forms


Auc probability

Pr(X = 2)

=.2109 × 1.0

AUC = probability!


Cumulative probability pr x x left tail

Cumulative Probability = Pr(X  x) = Left “Tail”

This figure illustrates Pr(X  2) on X ~b(4,.75)


Cumulative probability function

Pr(X =0) + Pr(X = 1)

Pr(X =0) + Pr(X = 1) + Pr(X = 2)

Pr(X =0) + Pr(X = 1) + … + Pr(X = 3)

Pr(X =0) + Pr(X = 1) + … + Pr(X = 4)

Cumulative Probability Function

Cumulative probability function (cdf) = cumulative probabilities for all outcome

Example: cdf for X~b(4, 0.75)

Pr(X  0) = 0.0039

Pr(X  1) = 0.0508

Pr(X  2) = 0.2617

Pr(X  3) = 0.6836

Pr(X  4) = 1.0000


Calculating binomial probabilities with the statable utility

Calculating Binomial Probabilities with the StaTable Utility

StaTable is a free computer program that calculates probabilitiesfor many types of random variables, including binomials


Statable binomial calculator

StaTable Binomial Calculator

Number of successes x

Binomial parameter p

Binomial parameter n

Calculates Pr(X = x)

Calculates Pr(X≤ x)


Statable probability calculator

StaTable Probability Calculator

StaTable

Exact and cumulative probability of “2” for X~b(n = 4, p = .75)

x = 2

p = .75

n = 4

Pr(X = 2) = .2109

Pr(X≤ 2) = .2617


6 5 expected value and variance for binomials

§6.5: Expected Value and Variance for Binomials

  • Expected value μ

  • Variance σ2

  • Shortcut formulas:


Expected value and variance binomials illustration

Expected Value and Variance, Binomials, Illustration

For X~b(4,.75)

μ = n∙p= (4)(.75) = 3

σ2 = n∙p∙q= (4)(.75)(.25) = 0.75


6 6 using the binomial

§6.6 Using the Binomial

  • Suppose we observe 2 successes in a “Four patients” experiment?

  • Assume X~b(4, .75)

  • 3 success are expected

  • Does the observation of 2 successes cast doubt on p = 0.75?

Pr(X  2) = 0.2617. What does this infer?


  • Login