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

## 1. BBA182 Applied Statistics Week 6 (2) Random variables – discrete random variables

DR SUSANNE HANSEN SARALEMAIL: [email protected]

HT TPS://PIAZZA.COM/CLASS/IXRJ5MMOX1U2T8?CID=4#

WWW.KHANACADEMY.ORG

DR SUSANNE HANSEN SARAL

1

## 2. Random Variables

Represent possible numerical values from a random experiments. Whichoutcome will occur, is not known, therefore the word “random”.

Random

Variables

Discrete

Random Variable

Continuous

Random Variable

DR SUSANNE HANSEN SARAL

Ch. 4-2

## 3. 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:

## 4. Discrete random variable

We can calculate the exact probability of a discrete random variable.Example: The probability of students coming late to class today out of all students

registered

Total students registered: 45

Students late for the class: 15

P(students late for the class) =

15

45

= .33 or 33 %

## 5. Continuous random variable

A random variable that has an unlimited set of values. Therefore called continuous random variableContinuous 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

## 6. 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:

## 7. Time, in seconds, it takes 110 employees to assemble a toaster

271 236 294 252 254 263 266 222 262 278 288262 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

263 242 288

255 294 268 255 272 271 291

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

269

276 248 299

263 247 252

261 266 269 236

DR SUSANNE HANSEN SARAL, [email protected]

## 8. Continuous random variable

The probability of a continuous variable is calculated in an interval (5 - 10), because the probability of aspecific continuous random variable is close to 0.

Example: The time it takes for each of 110 employees in a factory to assemble a toaster:

n = 110

1 employee assembles the toaster is 222 seconds out of 110

P(employee assembles the toaster in 222 seconds out of all 110 employees) =

this provides little useful information

1

110

= 0.009 or 0.9 %

## 9. 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

DR SUSANNE HANSEN SARAL, [email protected]

100 %

## 10. Probability Models

For both discrete and continuous variables, the collection of all possible outcomes (samplespace) 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

## 11.

Probability Model forDiscrete 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:

DR SUSANNE HANSEN SARAL

Ch. 4-11

## 12. Probability Model, also Probability Distributions Function, P(x) for Discrete Random Variables

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

4 possible outcomes

(values for x)

Probability Distribution

T

T

H

H

T

H

H

0

1/4 = .25

1

2/4 = .50

2

1/4 = .25

Probability

T

.50

.25

0

DR SUSANNE HANSEN SARAL

1

2

Ch. 4-12

x

## 13. 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:

x

0

1

2

3

Total

P(x)

0.1

0.2

0.4

0.3

1.00

DR SUSANNE HANSEN SARAL

13

## 14. Graphical illustration of the probability distribution of sandwich sales between 14:00 -16:00 hours

DR SUSANNE HANSEN SARAL14

## 15. Requirements for a probability distribution of a discrete random variable

1. 0 ≤ P(x) ≤ 1 for any value of x2. The individual probabilities of all outcomes sum up to 1;

P(x) 1

x

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.

DR SUSANNE HANSEN SARAL

Ch. 4-15

## 16. Cumulative Probability Function

(continued)Example: Toss 2 coins simultaneously

Let the random variable, X, be number of the heads. There are 4 possible

outcomes:

X Value

P(x)

F(x)

0

0.25

0.25

1 (x2)

0.50

0.75

2

0.25

1.00

F(x0) =