Chpt 5. Discrete Probability Distributions. 5-3 Mean, Variance, Standard Deviation, and Expectation

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1 Chpt 5 Discrete Probability Distributios 5-3 Mea, Variace, Stadard Deviatio, ad Expectatio 1/23

2 Homework p252 Applyig the Cocepts Exercises p /23

3 Objective Fid the mea, variace, stadard deviatio, ad expected value for a discrete radom variable. 3/23

4 Mea Objective: Studets will costruct a probability distributio for a radom discrete variable. The mea of a probability distributio is calculated like ay other mea. Idetical to weighted mea, simply ew termiology. Rather tha calculate mea by divisio of the umber of occurreces, the mea of a discrete probability distributio is foud usig the probability of each occurrece. I other words, the divisio has bee doe whe determiig the probabilities. 4 /23

5 Mea Objective: Studets will costruct a probability distributio for a radom discrete variable. Suppose we roll a die with the followig results. x What would be the average roll? f(x) We could list all the results oce for each time that result occurs: 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3,..., 6, 6, 6, 6 The we ca calculate the mea by addig up all the values ad dividig by the umber of values /23

6 or Objective: Studets will costruct a probability distributio for a radom discrete variable. We ca fid the mea value of a sigle roll by fidig the weighted mea. x f(x) Total 28 f (x i ) x i f (x i ) f (x i ) =, the umber of observatios Multiply the value of the roll (x) by the frequecy f(x): /23

7 Mea Objective: Studets will costruct a probability distributio for a radom discrete variable. The weighted mea of a discrete frequecy distributio is foud by µ = f (x i ) x i f (x i ) f (x i ) =, the umber of observatios We ca write the equatio for fidig the mea slightly differetly f (x ) x i i 7 /23

8 Mea Objective: Studets will costruct a probability distributio for a radom discrete variable. We ca write the equatio for fidig the mea slightly differetly by usig the defiitio of divisio ad applyig the distributive property. f (x ) x i i = 1 f (x ) x = i i f (x ) x i i = f (x ) i x X = p (x ) x i i i If we replace the frequecies with relative frequecies, the resultig table is a probability distributio. x p(x) f(x) /23

9 Mea Objective: Studets will costruct a probability distributio for a radom discrete variable. Now, to fid the mea value of a sigle roll usig the probability distributio we agai fid the weighted mea. The weightig this time are the probabilities. X = p (x i ) x i x p(x) = The average value of a sigle roll is /23

10 Mea Objective: Studets will costruct a probability distributio for a radom discrete variable. Suppose we roll a die ad use the theoretical probability for each roll. What would be the average roll? X P(x) To fid the mea value of a sigle roll we fid the weighted mea oce agai, this time usig the probability of each roll. µ = p (x i ) x i = The theoretical average value of a sigle roll is /23

11 Mea Objective: Studets will costruct a probability distributio for a radom discrete variable. Sice the probability is already a quotiet (just like relative frequecy) it is ot ecessary to divide. The mea of a discrete probability distributio is deoted μ (mu). μ = x1 P(x1) + x2 P(x2) x P(x) ( ) µ = p (x i ) x i 11/23

12 Expected Value Objective: Studets will costruct a probability distributio for a radom discrete variable. Aother Name for weighted mea The mea of a probability distributio is the expected value [E(x)] of a sigle evet. I the previous example the expected value of a sigle roll of a die is 3.5. The expected value, or expectatio, of a discrete radom variable of a probability distributio is the theoretical average of the variable. ( ) E (x ) = x i p (x i ) 12/23

13 Example Objective: Studets will costruct a probability distributio for a radom discrete variable. A store has the followig results for profits for each day; P & L (x) P(x) What is the expected value for daily profit? 13/23

14 Example Objective: Studets will costruct a probability distributio for a radom discrete variable. E (x ) = x i p (x i ) ( ) P & L (x) P(x) E (x ) = 1000(.4) (.3) (.2) (.1) = = 100 The expected value for daily profit would be +$ /23

15 Variace Objective: Studets will costruct a probability distributio for a radom discrete variable. The variace of a probability distributio is foud by multiplyig the squared deviatio by its correspodig probability, summig these products, ad dividig by the degrees of freedom. The formula for the variace of a frequecy distributio is s 2 = f i i (x i X ) 2 1 = 1 1 f i i (x i X ) 2, f i = frequecy of x i This ca also be rewritte for a probability distributio as (simply distribute): s 2 = 1 1 f i i (x i X ) 2 = f i 1 i (x i X ) 2 = p (x i ) i (x i X ) 2 This looks suspiciously like a weighted sum of squared deviatios or a expected squared deviatio. 15/23

16 Variace Objective: Studets will costruct a probability distributio for a radom discrete variable. We ca ow write a equatio for the populatio variace ( ) σ 2 = (x i µ) 2 P (x ) As before your book has a calculatioal formula σ 2 = x 2 i P (x i ) µ 2 Stadard Deviatio (σ) is simply the square root of the variace. ( ) σ = (x i µ) 2 P (x ) 16/23

17 Example Objective: Studets will costruct a probability distributio for a radom discrete variable. Our store has the followig results for profits (μ = $100); P & L (x) P(x) (x - 100) 2 (x - 100) 2 p(x) σ = (x µ) 2 P (x ) = i Our distributio of profits has a mea profit of $100 ad a stadard deviatio of $ /23

18 Example Objective: Studets will costruct a probability distributio for a radom discrete variable. Remember this? 3rd child 1st child p(b).51 p(g).49 2d child p(b).50 p(g).50 p(b).545 p(b).523 p(b).51 p(b).51 p(g).477 p(g).49 p(g).49 p(bbb) = =.1334 p(bbg) = =.1216 p(bgb) = =.1301 p(bgg) = =.1250 p(gbb) = =.1362 p(gbg) = =.1309 x P(x) p(g).455 p(b).46 p(g).54 p(ggb) = =.1026 p(ggg) = = /23

19 Example E (x ) = p (x i ) x i = = 1.53 Objective: Studets will costruct a probability distributio for a radom discrete variable. x P(x) σ = (x i µ) 2 P (x ) = (0 1.53) (1 1.53) (2 1.53) (3 1.53) I a 3 child family, we would expect 1.53 boys with a stadard deviatio of.8667 boys. 19/23

20 Example Objective: Studets will costruct a probability distributio for a radom discrete variable. A ski resort loses $70,000 per seaso whe it does ot sow very much ad makes $250,000 whe it sows a lot. The probability of it sowig at least 75 iches (i.e., a good seaso) is 40%. Fid the expected profit. Start with a table of the probability distributio. E (x ) = ( ) p (x i ) x i P & L (x) P(x) -$ $ = = $58000 The resort expects to profit a average $58000 per seaso. 20/23

21 Example Objective: Studets will costruct a probability distributio for a radom discrete variable. A talk radio show has four call-i phoe lies. Whe the radio show host is talkig to a caller other callers are put o hold. If all 4 lies are i use callers get a busy sigal. The probability that 0, 1, 2, 3, or 4 people will be placed o hold whe they call a radio talk show with four phoe lies is show i the distributio below. Fid the variace ad stadard deviatio for the data. x P(x) ( ) = = 1.59 E (x ) = x i p (x i ) ( ) σ 2 = (x i µ) 2 P (x ) (x - x) = ( 1.59) (.59) (.41) ( 1.41) ( 2.41) 2.04 = 1.26 σ = /23

22 Example Objective: Studets will costruct a probability distributio for a radom discrete variable. A talk radio show has four call-i phoe lies. Whe the radio show host is talkig to a caller other callers are put o hold. If all 4 lies are i use callers get a busy sigal. The probability that 0, 1, 2, 3, or 4 people will be placed o hold whe they call a radio talk show with four phoe lies is show i the distributio below. E (x ) = 1.59 σ x P(x) (x - x) Should the radio statio add additioal lies? Sice the expected umber of callers o hold is 1.59, ad the vast majority of time there are betwee 0 ad 3.8 callers o hold (±2 σ) there is o cost justificatio for addig more lies. 22/23

23 TI-84 Objective: Studets will costruct a probability distributio for a radom discrete variable. Now I will show you how you actually fid E(x) ad σ. x Eter x values i L1. P(x) Eter the probabilities for each x i L2. Stat Calc 1:1-Var Stats (L1, L2) Eter x = 1.59 σx = Remember: 4 decimals ad fial aswer with a setece! 23/23

A random variable is a variable whose value is a numerical outcome of a random phenomenon.

A random variable is a variable whose value is a numerical outcome of a random phenomenon. The Practice of Statistics, d ed ates, Moore, ad Stares Itroductio We are ofte more iterested i the umber of times a give outcome ca occur tha i the possible outcomes themselves For example, if we toss