4a probability concepts and binomial probability distributions
Download
1 / 22

4A: Probability Concepts and Binomial Probability Distributions - PowerPoint PPT Presentation


  • 186 Views
  • Uploaded on

4A: Probability Concepts and Binomial Probability Distributions. Definitions. Random variable  a numerical quantity that takes on different values depending on chance Population  the set of all possible values for a random variable

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about '4A: Probability Concepts and Binomial Probability Distributions' - lotus


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
4a probability concepts and binomial probability distributions

4A: Probability Concepts and Binomial Probability Distributions

Probability Concepts & Binomial Distributions


Definitions
Definitions Distributions

  • Random variable  a numerical quantity that takes on different values depending on chance

  • Population  the set of all possible values for a random variable

  • Event  an outcome or set of outcomes for a random variable

  • Probability  the proportion of times an event occurs in the population; (long-run) expected proportion

Probability Concepts & Binomial Distributions


Probability definition 1
Probability (Definition #1) Distributions

Probability is its relative frequency of the event in the population.

Example:

Let A  selecting a female at random from an HIV+ population

There are 600 people in the population.

There are 159 females.

Therefore, Pr(A) = 159 ÷ 600 = 0.265

Probability Concepts & Binomial Distributions


Probability definition 2
Probability (Definition #2) Distributions

Probability is the long run proportion when the process in repeated again and again under the same conditions.

  • Select 100 individuals at random

    • 24 are female

    • Pr(A)  24 ÷ 100 = 0.24

  • This is only an estimate (unless n is very very big)

Probability Concepts & Binomial Distributions


Probability definition 3
Probability (Definition #3) Distributions

Probability is a quantifiable level of belief between 0 and 1

Example: I believe a quarter of population is male. Therefore, in selecting individuals at random: Pr(male) ≈ 0.25

Probability Concepts & Binomial Distributions


Rules for probabilities
Rules for Probabilities Distributions

Probability Concepts & Binomial Distributions


Types of random variables
Types of Random Variables Distributions

  • Discrete have a finite set of possible outcomes,

    • e.g. number of females in a sample of size n (0, 1, 2, …, n)

    • We cover binomial random variables

  • Continuous have a continuum of possible outcomes

    • e.g., average body weight (lbs) in a sample (160, 160.5, 160.75, 160.825, …)

    • We cover Normal random variables

There are other random variable families, but only binomial (this lecture) and Normal (next lecture) families will be covered.

Probability Concepts & Binomial Distributions


Binomial random variables
Binomial random variables Distributions

  • Most popular type of discrete random variable

  • Bernoulli trial  random event characterized by “success” or “failure”

  • Examples

    • Coin flip (heads or tails)

    • Survival (yes or no)

Probability Concepts & Binomial Distributions


Binomial random variables cont
Binomial random variables (cont.) Distributions

  • Binomial random variable  random number of successes in n independent Bernoulli trials

  • A family of distributions identified by two parameters

    • n  number of trials

    • p  probability of success for each trial

  • Notation: X~b(n,p)

    • X  random variable

    • ~  “distributed as”

    • b(n, p)  binomial RV with parameters n and p

Probability Concepts & Binomial Distributions


Four patients example
“Four patients” example Distributions

  • A treatment is successful 75% of time

  • We treat 4 patients

  • X  random number of successes, which varies  0, 1, 2, 3, or 4 depending on binomial distribution X~b(4, 0.75)

Probability Concepts & Binomial Distributions


The binomial formula

The probability of Distributionsi successes is …

The Binomial Formula

Where

nCi= the binomial coefficient (next slide)

p = probability of success for each trial

q = probability of failure =1 – p

Probability Concepts & Binomial Distributions


Binomial coefficient choose function
Binomial Coefficient (“Choose Function”) Distributions

where

!  the factorial function: x! = x  (x – 1)  (x – 2)  …  1

Example: 4! = 4  3  2  1 = 24 By definition 1! = 1 and 0! = 1

nCi the number of ways to choose i items out of n

Example: “4 choose 2”:

Probability Concepts & Binomial Distributions


The four patients illustrative example
The “Four Patients” Illustrative Example Distributions

  • n = 4 and p = 0.75 (so q = 1 - 0.75 = 0.25)

  • Question: What is probability of 0 successes?  i = 0

  • Pr(X = 0) =nCi pi qn–i = 4C0 · 0.750 · 0.254–0= 1 · 1 · 0.0039 = 0.0039

Probability Concepts & Binomial Distributions


X b 4 0 75 continued
X Distributions~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

(Do not demonstrate all calculations. Students should prove to themselves they derive and interpret these values.)

Probability Concepts & Binomial Distributions


X b 4 0 75 continued1
X Distributions~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

Probability Concepts & Binomial Distributions


The probability mass function for x b 4 0 75
The Probability Mass Function for DistributionsX~b(4, 0.75)

Probability table for X~b(4,.75)

Probability curve for X~b(4,.75)

Probability Concepts & Binomial Distributions


Area under the curve auc

Get it? Distributions

Pr(X = 2) = .2109

Area Under The Curve (AUC)

The area under the curve (AUC) = probability!

Probability Concepts & Binomial Distributions


Cumulative probability left tail
Cumulative Probability (left tail) Distributions

  • Cumulative probability = Pr(X i) = probability less than or equal to i

  • Illustrative example: X~b(4, .75)

    • Pr(X  0) = Pr(X = 0) = .0039

    • Pr(X  1) = Pr(X  0) + Pr(X = 1) = .0039 + .0469 = 0.0508

    • Pr(X  2) = Pr(X  1) + Pr(X = 2) = .0508 + .2109 = 0.2617

    • Pr(X  3) = Pr(X  2) + Pr(X = 3) = .2617 + .4219 = 0.6836

    • Pr(X  4) = Pr(X  3) + Pr(X = 4) = .6836 + .3164 = 1.0000

Probability Concepts & Binomial Distributions


The cumulative mass function for x b 4 0 75
The Cumulative Mass Function for X~b(4, 0.75) Distributions

Probability Concepts & Binomial Distributions


Cumulative probability

Bring it on! Distributions

Cumulative Probability

Area under left tail = cumulative probability

Area under shaded bars in left tail sums to 0.2617:Pr(X  2) = 0.2617

Area under “curve” = probability

Probability Concepts & Binomial Distributions


Reasoning with probabilities
Reasoning with Probabilities Distributions

We use probability model to reasoning about uncertainty & chance.

I hypothesize p = 0.75, but observe only 2 successes. Should I doubt my hypothesis?

ANS: No. When p = 0.75, you’ll see 2 or fewer successes 25% of the time (not that unusual).

Probability Concepts & Binomial Distributions


Statable probability calculator
StaTable Probability Calculator Distributions

  • Three versions

    • Java (browser)

    • Windows

    • Palm

  • Calculates probabilities for many pmfs and pdfs

  • Example (right) is for a X~b(4,0.75) when x = 2

No of successes x

Pr(X = x)

Pr(X≤ x)

Probability Concepts & Binomial Distributions