Random variables – discrete random variables. Week 6 (2)

Сентябрь 14, 2022

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Random variables – discrete random variables. Week 6 (2)

Содержание

2. Random Variables Represent possible numerical values from a random experiments. Which outcome will occur, is not
3. Discrete random variable A discrete random variable is a possible outcome from a random experiment. It
4. Discrete random variable
5. Continuous random variable A random variable that has an unlimited set of values. Therefore called continuous
6. Continuous random variable A continuous random variable has an unlimited set of values. The probability of
7. Time, in seconds, it takes 110 employees to assemble a toaster 271 236 294 252 254
8. Continuous random variable
9. Employee assembly time in seconds Completion time (in seconds) Frequency Relative frequency % 220 – 229
10. Probability Models For both discrete and continuous variables, the collection of all possible outcomes (sample space)
11. Ch. 4- DR SUSANNE HANSEN SARAL Probability Model for Discrete Random Variables Let X be a
12. Ch. 4- DR SUSANNE HANSEN SARAL Probability Model, also Probability Distributions Function, P(x) for Discrete Random
13. Probability Distributions Function, P(x) for Discrete Random Variables (example) Sales of sandwiches in a sandwich shop:
14. Graphical illustration of the probability distribution of sandwich sales between 14:00 -16:00 hours DR SUSANNE HANSEN
15. Requirements for a probability distribution of a discrete random variable DR SUSANNE HANSEN SARAL Ch. 4-
16. Cumulative Probability Function DR SUSANNE HANSEN SARAL Ch. 4- (continued) X Value P(x) F(x) 0 0.25
17. DR SUSANNE HANSEN SARAL Ch. 4- (continued) x Value P(x) F(x0) A 0.18 0.18 = 0.18
18. Graphical illustration of P(x) DR SUSANNE HANSEN SARAL Ch. 4- Example: Let the random variable, X,
19. Graphical illustration of F(x0) Cumulative probability distribution, Ogive DR SUSANNE HANSEN SARAL Ch. 4- Example: Let
20. Cumulative Probability Function, F(x0) Practical application DR SUSANNE HANSEN SARAL The cumulative probability distribution can be
21. Cumulative Probability Function, F(x0) Practical application: Car dealer DR SUSANNE HANSEN SARAL The random variable, X,
22. Cumulative Probability Function, F(x0) Practical application DR SUSANNE HANSEN SARAL Example: If there are 3 cars
23. Cumulative Probability Function, F(x0) Practical application DR SUSANNE HANSEN SARAL Example: If only 2 cars are
24. 3/9/2017 DR SUSANNE HANSEN SARAL
25. Cumulative probability - solution DR SUSANNE HANSEN SARAL
26. Exercise In a geography exam the grade students obtained is the random variable X. It has
27. Cumulative probabilities - exercise Based on this, calculate the following: The cumulative probability distribution of X,
28. Properties of Discrete Random Variables The measurements of central tendency and variation for discrete random variables:
29. Expectations
30. Expected Value of a discrete random variable X: Example: Toss 2 coins, random variable, X =
31. Properties of Discrete Random Variables Expected Value (or mean) of a discrete random variable X: We
32. Expected value So, the expected value, E[X], of a discrete random variable is found by multiplying
33. Concept of expected value of a random variable A review of university textbooks reveals that 81
34. Expected value – calculation (example) Find the expected mean number of mistakes on pages: = (0)(.81)+(1)(.17)+(2)(.02)
35. Probability distribution of mistakes in textbooks DR SUSANNE HANSEN SARAL
36. Exercise A lottery offers 500 tickets for $ 3 each. If the biggest prize is $
37. Exercise A lottery offers 500 tickets for $ 3 each. If the biggest prize is $
39. Скачать презентацию

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Random Variables
Represent possible numerical values from a random experiments. Which outcome

will occur, is not known, therefore the word “random”.

DR SUSANNE HANSEN SARAL

Ch. 4-

Random
Variables

Discrete
Random Variable

Continuous
Random Variable

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Discrete random variable
A discrete random variable is a possible outcome

from a random experiment.
It takes on countable values, integers.
Examples of discrete random variables:
Number of cars crossing the Bosphorus Bridge every day
Number of journal subscriptions
Number of visits on a given homepage per day
We can calculate the exact probability of a discrete random variable:

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Discrete random variable

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Continuous random variable
A random variable that has an unlimited set of

values. Therefore called continuous random variable
Continuous random variables are common in business applications for modeling physical
quantities such as height, volume and weight, and monetary quantities such as profits, revenues
and expenses.
Examples:
The weight of cereal boxes filled by a filling machine in grams
Air temperature on a given summer day in degrees Celsius
Height of a building in meters
Annual profits in $ of 10 Turkish companies

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Continuous random variable
A continuous random variable has an unlimited set of

values.
The probability of a continuous variable is calculated in an interval (ex.: 5 -10), because the probability of a specific continuous random variable is close to 0. This would not provide useful information.
Example: The time it takes for each of 110 employees in a factory to assemble a toaster:

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Time, in seconds, it takes 110 employees to assemble a

toaster

271 236 294 252 254 263 266 222 262 278 288
262 237 247 282 224 263 267 254 271 278 263
262 288 247 252 264 263 247 225 281 279 238
252 242 248 263 255 294 268 255 272 271 291
263 242 288 252 226 263 269 227 273 281 267
263 244 249 252 256 263 252 261 245 252 294
288 245 251 269 256 264 252 232 275 284 252
263 274 252 252 256 254 269 234 285 275 263
263 246 294 252 231 265 269 235 275 288 294
263 247 252 269 261 266 269 236 276 248 299

DR SUSANNE HANSEN SARAL, SUSANNE.SARAL@GMAIL.COM

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Continuous random variable

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Employee assembly time in seconds
Completion time (in seconds) Frequency Relative frequency

%
220 – 229 5 4.5
230 – 239 8 7.3
240 – 249 13 11.8
250 – 259 22 20.0
260 – 269 32 29.1
270 – 279 13 11.8
280 – 289 10 9.1
290 – 300 7 6.4
Total 110 100 %

DR SUSANNE HANSEN SARAL, SUSANNE.SARAL@GMAIL.COM

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Probability Models
For both discrete and continuous variables, the collection of

all possible outcomes (sample space) and probabilities associated with them is called the probability model.
For a discrete random variable, we can list the probability of all possible values in a table.
For example, to model the possible outcomes of a dice, we let X be the random variable called the “number showing on the face of the dice”. The probability model for X is therefore:
1/6 if x = 1, 2, 3, 4, 5, or 6
P(X = x) =
0 otherwise

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Ch. 4-
DR SUSANNE HANSEN SARAL
Probability Model for
Discrete Random

Variables

Let X be a discrete random variable and x be one of its possible values
The probability that random variable X takes specific value x is denoted P(X = x).
In the dice example: X is the random variable “the number showing on the
dice” and it’s value, x = the specific number. Ex.: P( X = 3)
The probability distribution function, P(x) of a random variable, X, is a representation of the probabilities for all the possible outcomes, x.
The function can be shown algebraically, graphically, or with a table:

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Ch. 4-
DR SUSANNE HANSEN SARAL
Probability Model, also Probability Distributions Function,

P(x) for Discrete Random Variables

0 1/4 = .25
1 2/4 = .50
2 1/4 = .25

Experiment: Toss 2 Coins simultaneously. Let the random variable, X, be the # heads

4 possible outcomes (values for x)

Probability Distribution

0 1 2 x

.50
.25

Probability

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Probability Distributions Function, P(x) for Discrete Random Variables (example)
Sales of

sandwiches in a sandwich shop:
Let, the random variable X, represent the number of sandwiches sold within the time period of 14:00 - 16:00 hours in one given day. The probability distribution function, P(x) of sales is given by the table here below:

DR SUSANNE HANSEN SARAL

Слайд 14

Graphical illustration of the probability distribution of sandwich sales between

14:00 -16:00 hours

DR SUSANNE HANSEN SARAL

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Requirements for a probability distribution of a discrete random variable
DR

SUSANNE HANSEN SARAL

Ch. 4-

1. 0 ≤ P(x) ≤ 1 for any value of x
2. The individual probabilities of all outcomes sum up to 1;
All possible values of X are mutually exclusive and collectively exhaustive (the outcomes make up the entire sample space), therefore the probabilities for these events must sum to 1.

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Cumulative Probability Function

DR SUSANNE HANSEN SARAL
Ch. 4-
(continued)
X Value P(x) F(x)

0 0.25 0.25
1 (x2) 0.50 0.75
2 0.25 1.00

Example: Toss 2 coins simultaneously
Let the random variable, X, be number of the heads. There are 4 possible outcomes:

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DR SUSANNE HANSEN SARAL
Ch. 4-
(continued)
x Value P(x) F(x0)
A 0.18

0.18 = 0.18
B 0.32 0.50 = 0.18 + 0.32
C 0.25 0.75 = 0.50 + 0.25
D 0.07 0.82 = 0.75 + 0.07
E 0.03 0.85 = 0.82 + 0.03
F 0.15 1.00 = 0.85 + 0.15

Example: Let the random variable, X, be the grades obtained in a geography exam and x = A, B, C, D, E, F are the possible outcomes/values :

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Graphical illustration of P(x)

DR SUSANNE HANSEN SARAL
Ch. 4-
Example: Let the

random variable, X, be the grades obtained in a geography exam.

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Graphical illustration of F(x0) Cumulative probability distribution, Ogive
DR SUSANNE HANSEN

SARAL

Ch. 4-

Example: Let the random variable, X, be the grades obtained in a geography exam.

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Cumulative Probability Function, F(x0) Practical application
DR SUSANNE HANSEN SARAL
The cumulative

probability distribution can be used for example for inventory planning?
Based on an analysis of it’s sales history, the manager of a car dealer knows that on any single day the number of Toyota cars sold can vary from 0 to 5.

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Cumulative Probability Function, F(x0) Practical application: Car dealer
DR SUSANNE HANSEN

SARAL

The random variable, X, is the number of possible cars sold in a day:

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Cumulative Probability Function, F(x0) Practical application
DR SUSANNE HANSEN SARAL
Example: If

there are 3 cars in stock. The car dealer will be able to satisfy 85% of the customers

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Cumulative Probability Function, F(x0) Practical application
DR SUSANNE HANSEN SARAL
Example: If

only 2 cars are in stock, then 35 % [(1-.65) x 100]
of the customers will not have their needs satisfied.

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3/9/2017
DR SUSANNE HANSEN SARAL

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Cumulative probability - solution

DR SUSANNE HANSEN SARAL

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Exercise
In a geography exam the grade students obtained is the random

variable X. It has been found that students have the following probabilities of getting a specific grade:
A: .18 D: .07
B: .32 E: .03
C: .25 F: .15
Based on this, calculate the following:
The cumulative probability distribution of X, F(x)
The probability of getting a higher grade than B
The probability of getting a lower grade than C
The probability of getting a grade higher than D
The probability of getting a lower grade than B

DR SUSANNE HANSEN SARAL

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Cumulative probabilities - exercise
Based on this, calculate the following:
The cumulative probability

distribution of X, F(x0)
The probability of getting a higher grade than B, P(x > B)
The probability of getting a lower grade than C, P(x < C)
The probability of getting a grade higher than D P(x > D)
The probability of getting a lower grade than B P(x < B)

DR SUSANNE HANSEN SARAL

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Properties of Discrete Random Variables
The measurements of central tendency and variation

for discrete random variables:
Expected value E[X] of a discrete random variable - expectations
Expected Variance of a discrete random variable
Expected Standard deviation of a discrete random variable
Why do we refer to expected value?

DR SUSANNE HANSEN SARAL

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Expectations

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Expected Value of a discrete random variable X:
Example:
Toss 2 coins,

random variable, X = # of heads,
(TT, HT,TH,HH) compute the expected value of X:

DR SUSANNE HANSEN SARAL

Ch. 4-

x P(x)
0 .25
1 .50
2 .25

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Properties of Discrete Random Variables

Expected Value (or mean)

of a discrete random variable X:
We weigh the possible outcomes by
the probabilities of their occurrence:
E(x) = (0 x .25) + (1 x .50) + (2 x .25) = 1.0

DR SUSANNE HANSEN SARAL

Ch. 4-

x P(x)
0 .25
1 .50
2 .25

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Expected value
So, the expected value, E[X], of a discrete random

variable is found by multiplying each possible value of the random variable by the probability that it occurs and then summing all the products:
The expected value of tossing two coins simultaneously is therefore:

E[x] = (0 x .25) + (1 x .50) + (2 x .25) = 1.0

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Concept of expected value of a random variable
A review of university

textbooks reveals that 81 % of the pages have no mistakes, 17 % of
the pages have one mistake and 2% have two mistakes.
We use the random variable X to denote the number of mistakes on a page chosen at random
from a textbook with possible values, x, of 0, 1 and 2 mistakes.
With a probability distribution of :
P(0) = .81 P(1) = .17 P(2) = .02
How do we calculate the expected value(average) of mistakes per page?

DR SUSANNE HANSEN SARAL

Слайд 34

Expected value – calculation (example)
Find the expected mean number of mistakes

on pages:
= (0)(.81)+(1)(.17)+(2)(.02) =.21
From this result we can conclude that over a large number of pages, the expectation would be to find an average of 21 % mistakes per page in business textbooks.

DR SUSANNE HANSEN SARAL

Слайд 35

Probability distribution of mistakes in textbooks
DR SUSANNE HANSEN SARAL

Слайд 36

Exercise
A lottery offers 500 tickets for $ 3 each. If

the biggest prize is $ 250 and 4 second prizes are $ 50 each :
a) What are the possible outcomes?
b) What is the expected value, E[X], of a single ticket?
c) Now, include the cost of the ticket you bought. What is the expected value now?
d) Knowing the value calculated in part b) does it make sense to buy a lottery ticket?
e) What is the expected value the lottery company can expect to gain from the lottery
sale?

Слайд 37

Exercise
A lottery offers 500 tickets for $ 3 each. If

the biggest prize is $ 250 and 4 second prizes are $ 50 each.
a) What are the possible outcomes? Winning the large prize of $ 250, 1/500, winning one of the 4
prizes of $ 50, 4/500 and winning nothing, $ 0, 495/500
b) What is the expected value, E[X], of a single ticket?
E[X] = $ 250 x (1/500) + $ 50x (4/500) + $ 0 x (495/500) = $ 0.50 +$ 0.40 +$ 0.00 = $ 0.90
c) Now, include the cost of the ticket you bought. What is the expected value now?
E[X] = $ 0.90 - $ 3.00 = $ - 2.10
d) Knowing the value calculated in part b) does it make sense to buy a lottery ticket?

Random variables – discrete random variables. Week 6 (2)

Содержание

Random VariablesRepresent possible numerical values from a random experiments. Which outcome

Discrete random variableA discrete random variable is a possible outcome

Discrete random variable

Continuous random variableA random variable that has an unlimited set of

Continuous random variableA continuous random variable has an unlimited set of

Time, in seconds, it takes 110 employees to assemble a

Continuous random variable

Employee assembly time in seconds Completion time (in seconds) Frequency Relative frequency

Probability ModelsFor both discrete and continuous variables, the collection of

Ch. 4-DR SUSANNE HANSEN SARAL Probability Model for Discrete Random

Ch. 4-DR SUSANNE HANSEN SARAL Probability Model, also Probability Distributions Function,

Probability Distributions Function, P(x) for Discrete Random Variables (example) Sales of

Graphical illustration of the probability distribution of sandwich sales between

Requirements for a probability distribution of a discrete random variableDR

Cumulative Probability Function DR SUSANNE HANSEN SARALCh. 4-(continued)X Value P(x) F(x)

DR SUSANNE HANSEN SARALCh. 4-(continued)x Value P(x) F(x0) A 0.18

Graphical illustration of P(x) DR SUSANNE HANSEN SARALCh. 4-Example: Let the

Graphical illustration of F(x0) Cumulative probability distribution, OgiveDR SUSANNE HANSEN

Cumulative Probability Function, F(x0) Practical applicationDR SUSANNE HANSEN SARALThe cumulative

Cumulative Probability Function, F(x0) Practical application: Car dealerDR SUSANNE HANSEN

Cumulative Probability Function, F(x0) Practical applicationDR SUSANNE HANSEN SARALExample: If

Cumulative Probability Function, F(x0) Practical applicationDR SUSANNE HANSEN SARALExample: If

3/9/2017DR SUSANNE HANSEN SARAL

Cumulative probability - solution DR SUSANNE HANSEN SARAL

ExerciseIn a geography exam the grade students obtained is the random

Cumulative probabilities - exerciseBased on this, calculate the following:The cumulative probability

Properties of Discrete Random VariablesThe measurements of central tendency and variation

Expectations

Expected Value of a discrete random variable X:Example:Toss 2 coins,

Properties of Discrete Random Variables Expected Value (or mean)

Expected valueSo, the expected value, E[X], of a discrete random

Concept of expected value of a random variable A review of university

Expected value – calculation (example)Find the expected mean number of mistakes

Probability distribution of mistakes in textbooksDR SUSANNE HANSEN SARAL

ExerciseA lottery offers 500 tickets for $ 3 each. If

ExerciseA lottery offers 500 tickets for $ 3 each. If

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