Discrete random variables – expected variance and standard deviation. Discrete Probability Distributions. Week 7 (1)

Сентябрь 14, 2022

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Discrete random variables – expected variance and standard deviation. Discrete Probability Distributions. Week 7 (1)

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2. DR SUSANNE HANSEN SARAL
3. Cumulative Probability Function, F(x0) Practical application: Car dealer DR SUSANNE HANSEN SARAL The random variable, X,
4. Cumulative Probability Function, F(x0) Practical application DR SUSANNE HANSEN SARAL Example: If there are 3 cars
5. Cumulative Probability Function, F(x0) Practical application DR SUSANNE HANSEN SARAL Example: If only 2 cars are
6. Properties of discrete random variables: Expected value E[x] = (0 x .25) + (1 x .50)
7. Expected value for a discrete random variable Exercise X is a discrete random variable. The graph
8. Expected value for a discrete random variable X is a discrete random variable. The graph below
9. Expected variance of a Discrete Random Variables DR SUSANNE HANSEN SARAL
10. Variance of a discrete random variable DR SUSANNE HANSEN SARAL
11. Variance and Standard Deviation Ch. 4- DR SUSANNE HANSEN SARAL
12. At a car dealer the number of cars sold daily could vary between 0 and 5
13. Calculation of variance of discrete random variable. Car sales – example DR SUSANNE HANSEN SARAL
14. Class exercise A car dealer calculates the proportion of new cars sold that have been returned
15. Dan’s computer Works – class exercise The number of computers sold per day at Dan’s Computer
16. Dan’s computer Works – class exercise The number of computers sold per day at Dan’s Computer
17. Dan’s computer Works – class exercise The number of computers sold per day at Dan’s Computer
18. Dan’s computer Works – class exercise DR SUSANNE HANSEN SARAL
19. Quizz A small school employs 5 teachers who make between $40,000 and $70,000 per year. One
20. Khan Academy – Empirical Rule A company produces batteries with a mean life time of 1’300
21. Stating that two events are statistically independent means that the probability of one event occurring is
22. The time it takes a car to drive from Istanbul to Sinop is an example of
23. Probability is a numerical measure about the likelihood that an event will occur. TRUE FALSE
24. Suppose that you enter a lottery by obtaining one of 20 tickets that have been distributed.
25. If we flip a coin three times, the probability of getting three heads is 0.125. TRUE
26. The number of products bought at a local store is an example of a discrete random
27. Empirical rule – Khan Academy a) Which shape does a distribution need to have to apply
28. Probability Distributions Continuous Probability Distributions Binomial Probability Distributions Discrete Probability Distributions Uniform Normal Exponential DR SUSANNE
29. Binomial Probability Distribution Bi-nominal (from Latin) means: Two-names A fixed number of observations, n e.g., 15
30. Possible Binomial Distribution examples A manufacturing plant labels products as either defective or acceptable A firm
31. The Binomial Distribution The binomial distribution is used to find the probability of a specific or
32. The Binomial Distribution The binomial formula is: 2 – The symbol ! means factorial, and n!
33. Example: Calculating a Binomial Probability What is the probability of one success in five observations if
34. Binomial probability - Calculating binomial probabilities Suppose that Ali, a real estate agent, has 5 people
35. Solving Problems with the Binomial Formula Find the probability of 4 people buying a house out
36. Class exerise Find the probability of 3 people buying a house out of 5 people, when
37. P( X = 3) ? Find the probability of 3 people buying a house out of
38. Creating a probability distribution with the Binomial Formula – house sale example 2 – TABLE 2.8
39. Binomial Probability Distribution house sale example n = 5, P= .4 DR SUSANNE HANSEN SARAL
40. DR SUSANNE HANSEN SARAL
41. DR SUSANNE HANSEN SARAL
42. Shape of Binomial Distribution The shape of the binomial distribution depends on the values of P
43. Binomial Distribution shapes When P = .5 the shape of the distribution is perfectly symmetrical and
44. Using Binomial Tables instead of to calculating Binomial probabilites DR SUSANNE HANSEN SARAL Ch. 4- Examples:
45. Solving Problems with Binomial Tables MSA Electronics is experimenting with the manufacture of a new USB-stick
47. Скачать презентацию

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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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Properties of discrete random variables: Expected value

E[x] = (0

x .25) + (1 x .50) + (2 x .25) = 1.0

DR SUSANNE HANSEN SARAL

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Expected value for a discrete random variable Exercise
X is a discrete

random variable. The graph below defines a probability distribution, P(X) for X.
What is the expected value of X?

DR SUSANNE HANSEN SARAL

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Expected value for a discrete random variable
X is a discrete random

variable. The graph below defines a probability distribution, P(X) for X.
What is the expected value of X?

DR SUSANNE HANSEN SARAL

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Expected variance of a Discrete Random Variables

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Variance of a discrete random variable

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Variance and Standard Deviation
Ch. 4-
DR SUSANNE HANSEN SARAL

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At a car dealer the number of cars sold daily could

vary between 0 and 5 cars, with the probabilities given in the table. Find the expected value and variance for this probability distribution

DR SUSANNE HANSEN SARAL

Ch. 4-

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Calculation of variance of discrete random variable. Car sales –

example

DR SUSANNE HANSEN SARAL

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Class exercise
A car dealer calculates the proportion of new cars

sold that have been returned a various number of times for the correction of defects during the guarantee period. The results are as follows:
Graph the probability distribution function
Calculate the cumulative probability distribution
What is the probability that cars will be returned for corrections more than two times? P(x > 2)
P(x < 2)?
Find the expected value of the number of a car for corrections for defects during the guarantee period
Find the expected variance

DR SUSANNE HANSEN SARAL

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Dan’s computer Works – class exercise
The number of computers sold

per day at Dan’s Computer Works is defined by the following probability distribution:
Calculate the expected value of number of computer sold per day:

DR SUSANNE HANSEN SARAL

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Dan’s computer Works – class exercise
The number of computers sold

per day at Dan’s Computer Works is defined by the following probability distribution:
Calculate the expected value of number of computer sold per day:
E[x]= (0 x 0.05) + (1 x 0.1) + (2 x 0.2) + (3 x 0.2) + (4 x 0.2) + (5 x 0.15) + (6 x 0.1) = 3.25 rounded to 3

DR SUSANNE HANSEN SARAL

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Dan’s computer Works – class exercise
The number of computers sold

per day at Dan’s Computer Works is defined by the following probability distribution:
Calculate the variance of number of computer sold per day:

DR SUSANNE HANSEN SARAL

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Dan’s computer Works – class exercise

DR SUSANNE HANSEN SARAL

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Quizz
A small school employs 5 teachers who make between $40,000 and $70,000

per year.
One of the 5 teachers, Valerie, decides to teach part-time which decreases her salary from $40,000 to $20,000 per year. The rest of the salaries stay the same.
How will decreasing Valerie's salary affect the mean and median?
Please choose from one of the following options:
A) Both the mean and median will decrease.
B) The mean will decrease, and the median will stay the same.
C)The median will decrease, and the mean will stay the same.
D) The mean will decrease, and the median will increase.

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Khan Academy – Empirical Rule
A company produces batteries with a

mean life time of 1’300 hours and a standard deviation of 50 hours. Use the Empirical rule (68 – 95 – 99.7 %) to estimate the probability of a battery to have a lifetime longer than 1’150 hours. P (x > 1’150 hours)
Which of the following is the right answer?
95 %
84%
73%
99.85%

DR SUSANNE HANSEN SARAL

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Stating that two events are statistically independent means that the probability

of one event occurring is independent of the probability of the other event having occurred.

TRUE
FALSE

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The time it takes a car to drive from Istanbul to

Sinop is an example of a discrete random variable
True
False

DR SUSANNE HANSEN SARAL

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Probability is a numerical measure about the likelihood that an event

will occur.

TRUE
FALSE

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Suppose that you enter a lottery by obtaining one of 20

tickets that have been distributed. By using the relative frequency method, you can determine that the probability of your winning the lottery is 0.15.
TRUE
FALSE

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If we flip a coin three times, the probability of getting

three heads is 0.125.

TRUE
FALSE

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The number of products bought at a local store is an

example of a discrete random variable.

TRUE
FALSE

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Empirical rule – Khan Academy
a) Which shape does a

distribution need to have to apply the Empirical Rule?
b) The lifespans of zebras in a particular zoo are normally distributed. The average zebra lives 20.5 years, the standard deviation is 3.9, years.
Use the empirical rule (68-95-99.7%) to estimate the probability of a zebra living less than 32.2 years.

DR SUSANNE HANSEN SARAL

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Probability Distributions
Continuous
Probability Distributions
Binomial
Probability Distributions
Discrete
Probability Distributions
Uniform
Normal
Exponential
DR SUSANNE HANSEN SARAL
Ch.

Poisson

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Binomial Probability Distribution Bi-nominal (from Latin) means: Two-names
A fixed number of observations,

n
e.g., 15 tosses of a coin; ten light bulbs taken from a warehouse
Only two mutually exclusive and collectively exhaustive possible outcomes
e.g., head or tail in each toss of a coin; defective or not defective light bulb
Generally called “success” and “failure”
Probability of success is P , probability of failure is 1 – P
Constant probability for each observation
e.g., Probability of getting a tail is the same each time we toss the coin
Observations are independent
The outcome of one observation does not affect the outcome of the other

DR SUSANNE HANSEN SARAL

Ch. 4-

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Possible Binomial Distribution examples
A manufacturing plant labels products as either

defective or acceptable
A firm bidding for contracts will either get a contract or not
A marketing research firm receives survey responses of “yes I will buy” or “no I
will not”
New job applicants either accept the offer or reject it
A customer enters a store will either buy a product or will not buy a product

DR SUSANNE HANSEN SARAL

Ch. 4-

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The Binomial Distribution
The binomial distribution is used to find the probability

of a specific or cumulative number of successes in n trials

2 –

DR SUSANNE HANSEN SARAL

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The Binomial Distribution
The binomial formula is:
2 –
The symbol ! means

factorial, and n! = n(n – 1)(n – 2)…(1)
4! = (4)(3)(2)(1) = 24

Also, 1! = 1 and 0! = 0 by definition

DR SUSANNE HANSEN SARAL

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Example: Calculating a Binomial Probability
What is the probability of one success

in five observations if the probability of success is 0.1?
x = 1, n = 5, and P = 0.1

DR SUSANNE HANSEN SARAL

Ch. 4-

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Binomial probability - Calculating binomial probabilities
Suppose that Ali, a real

estate agent, has 5 people interested in buying a house in the area Ali’s real estate agent operates.
Out of the 5 people interested how many people will actually buy a house if the probability of selling a house is 0.40. P(X = 4)?

DR SUSANNE HANSEN SARAL

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Solving Problems with the Binomial Formula
Find the probability of 4

people buying a house out of 5 people, when the probability of success is .40

2 –

n = 5, r = 4, p = 0.4, and q = 1 – 0.4 = 0.6

DR SUSANNE HANSEN SARAL

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Class exerise
Find the probability of 3 people buying a house

out of 5 people, when the probability of success is .40
P(X =3) ?
n = 5, r = 3, p = 0.4, and q = 1 – 0.4 = 0.6

DR SUSANNE HANSEN SARAL

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P( X = 3) ?
Find the probability of 3

people buying a house out of 5 people, when the probability of success is .40
n = 5, r = 3, p = 0.4, and q = 1 – 0.4 = 0.6

DR SUSANNE HANSEN SARAL

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Creating a probability distribution with the Binomial Formula – house

sale example

2 –

TABLE 2.8 – Binomial Distribution
for n = 5, p = 0.40

DR SUSANNE HANSEN SARAL

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Binomial Probability Distribution house sale example n = 5, P=

DR SUSANNE HANSEN SARAL

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DR SUSANNE HANSEN SARAL

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DR SUSANNE HANSEN SARAL

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Shape of Binomial Distribution
The shape of the binomial distribution depends on

the values of P and n

n = 5 P = 0.1

n = 5 P = 0.5

Mean

P(x)

Here, n = 5 and P = 0.1

Here, n = 5 and P = 0.5

DR SUSANNE HANSEN SARAL

Ch. 4-

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Binomial Distribution shapes
When P = .5 the shape of the

distribution is perfectly symmetrical and resembles a bell-shaped (normal distribution)
When P = .2 the distribution is skewed right. This skewness increases as P becomes smaller.
When P = .8, the distribution is skewed left. As P comes closer to 1, the amount of skewness increases.

DR SUSANNE HANSEN SARAL

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Using Binomial Tables instead of to calculating Binomial probabilites
DR SUSANNE

HANSEN SARAL

Ch. 4-

Examples:
n = 10, x = 3, P = 0.35: P(x = 3|n =10, p = 0.35) = .2522
n = 10, x = 8, P = 0.45: P(x = 8|n =10, p = 0.45) = .0229

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Solving Problems with Binomial Tables
MSA Electronics is experimenting with the manufacture

of a new USB-stick and is looking into the
Every hour a random sample of 5 USB-sticks is taken
The probability of one USB-stick being defective is 0.15
What is the probability of finding 3, 4, or 5 defective USB-sticks ?
P( x = 3), P(x = 4 ), P(x= 5)

2 –

n = 5, p = 0.15, and r = 3, 4, or 5

DR SUSANNE HANSEN SARAL

Discrete random variables – expected variance and standard deviation. Discrete Probability Distributions. Week 7 (1)

Содержание

DR SUSANNE HANSEN SARAL

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

Properties of discrete random variables: Expected value E[x] = (0

Expected value for a discrete random variable ExerciseX is a discrete

Expected value for a discrete random variableX is a discrete random

Expected variance of a Discrete Random Variables DR SUSANNE HANSEN SARAL

Variance of a discrete random variable DR SUSANNE HANSEN SARAL

Variance and Standard DeviationCh. 4-DR SUSANNE HANSEN SARAL

At a car dealer the number of cars sold daily could

Calculation of variance of discrete random variable. Car sales –

Class exerciseA car dealer calculates the proportion of new cars

Dan’s computer Works – class exerciseThe number of computers sold

Dan’s computer Works – class exerciseThe number of computers sold

Dan’s computer Works – class exerciseThe number of computers sold

Dan’s computer Works – class exercise DR SUSANNE HANSEN SARAL

QuizzA small school employs 5 teachers who make between $40,000 and $70,000

Khan Academy – Empirical RuleA company produces batteries with a

Stating that two events are statistically independent means that the probability

The time it takes a car to drive from Istanbul to

Probability is a numerical measure about the likelihood that an event

Suppose that you enter a lottery by obtaining one of 20

If we flip a coin three times, the probability of getting

The number of products bought at a local store is an

Empirical rule – Khan Academy a) Which shape does a

Probability DistributionsContinuous Probability DistributionsBinomialProbability DistributionsDiscrete Probability DistributionsUniformNormalExponentialDR SUSANNE HANSEN SARALCh.

Binomial Probability Distribution Bi-nominal (from Latin) means: Two-namesA fixed number of observations,

Possible Binomial Distribution examples A manufacturing plant labels products as either

The Binomial DistributionThe binomial distribution is used to find the probability

The Binomial DistributionThe binomial formula is:2 – The symbol ! means

Example: Calculating a Binomial ProbabilityWhat is the probability of one success

Binomial probability - Calculating binomial probabilitiesSuppose that Ali, a real

Solving Problems with the Binomial FormulaFind the probability of 4

Class exeriseFind the probability of 3 people buying a house

P( X = 3) ? Find the probability of 3

Creating a probability distribution with the Binomial Formula – house

Binomial Probability Distribution house sale example n = 5, P=

DR SUSANNE HANSEN SARAL

DR SUSANNE HANSEN SARAL

Shape of Binomial DistributionThe shape of the binomial distribution depends on

Binomial Distribution shapes When P = .5 the shape of the

Using Binomial Tables instead of to calculating Binomial probabilitesDR SUSANNE

Solving Problems with Binomial TablesMSA Electronics is experimenting with the manufacture

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