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Презентации » Математика » Random variables – discrete random variables. Week 6 (2)
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Слайды и текст к этой презентации:
№3 слайд
Содержание слайда: 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 слайд
Содержание слайда: 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) = = .33 or 33 %
№5 слайд
Содержание слайда: 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
№6 слайд
Содержание слайда: 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 слайд
Содержание слайда: 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
№8 слайд
Содержание слайда: Continuous random variable
The probability of a continuous variable is calculated in an interval (5 - 10), because the probability of a specific 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) = = 0.009 or 0.9 %
this provides little useful information
№10 слайд
Содержание слайда: 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
№13 слайд
Содержание слайда: 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:
№26 слайд
Содержание слайда: 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
№27 слайд
Содержание слайда: 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)
№28 слайд
Содержание слайда: 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?
№29 слайд
Содержание слайда: Expectations
Of course, we cannot predict exactly which number will occur when we roll a dice, but we can say what we expect to happen on average, in the long run (therefore the name expectation)
The expected value of rolling a dice infinitively is a parameter of the probability model. In fact, it is the mean,
We’ll write it as: E[X], for Expected value
The expected value is not an average of data values, but calculated from the probability distribution of rolling one dice infinitively.
№32 слайд
Содержание слайда: 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:
№33 слайд
Содержание слайда: 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?
№34 слайд
Содержание слайда: 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.
№36 слайд
Содержание слайда: 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?
№37 слайд
Содержание слайда: 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?
№38 слайд
Содержание слайда: Exercise
Although no single person will lose $ 2.10, because they either lose $3 or win $ 250 or $ 50.
$ 2.10 is the amount in average that the lottery organization gains per ticket.
The lottery can therefore expect to make 500 x $ 2.10 = $ 1050 by selling 500 lottery tickets.
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