Презентация Discrete random variables – expected variance and standard deviation. Discrete Probability Distributions. Week 7 (1) онлайн
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Слайды и текст к этой презентации:
№1 слайд
Содержание слайда: BBA182 Applied Statistics
Week 7 (1)Discrete random variables – expected variance and standard deviation
Discrete Probability Distributions
Dr Susanne Hansen Saral
Email: susanne.saral@okan.edu.tr
https://piazza.com/class/ixrj5mmox1u2t8?cid=4#
www.khanacademy.org
№2 слайд
Содержание слайда: Cumulative Probability Function, F()
Practical application
№3 слайд
Содержание слайда: Cumulative Probability Function, F(x0)
Practical application: Car dealer
№4 слайд
Содержание слайда: Cumulative Probability Function, F(x0)
Practical application
№5 слайд
Содержание слайда: Cumulative Probability Function, F(x0)
Practical application
№6 слайд
Содержание слайда: Properties of discrete random variables:
Expected value
The expected value, E[X], also called the mean, 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 :
№7 слайд
Содержание слайда: 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?
№8 слайд
Содержание слайда: 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?
№9 слайд
Содержание слайда: Expected variance
of a 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
№10 слайд
Содержание слайда: Variance of a discrete random variable
The variance is the measure of the spread of a set of numerical observations to the expected value, E[X].
For a discrete random variable we define the variance as the weighted average of the squares of its possible deviations (x - ):
№11 слайд
Содержание слайда: Variance and Standard Deviation
Let X be a discrete random variable. The expectation of the average of squared
deviations about the mean, , is called the expected variance, denoted and given
by:
Expected Standard Deviation of a discrete random variable X
№12 слайд
Содержание слайда: Exercise:
n n Expected value,E[X], and variance, of car sales
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
№13 слайд
Содержание слайда: Calculation of variance of discrete random variable.
Car sales – example
Calculating the expected value:
E(x) = (0)(.15)+(1)(.3)+(2)(.2)+(3)(.2)+(4)(.1)+(5)(.05)= 1.95 rounded up to 2 (discrete random variable)
Calculating the expected variance:
=
№14 слайд
Содержание слайда: 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
№15 слайд
Содержание слайда: 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:
№16 слайд
Содержание слайда: 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
№17 слайд
Содержание слайда: 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:
№18 слайд
Содержание слайда: 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:
=(0.05) +(0.1)+ + (0.2)+
(0.2)+ (0.15) + (0.1) = 2.69
= 2.69
№19 слайд
Содержание слайда: 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.
№20 слайд
Содержание слайда: 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%
№21 слайд
Содержание слайда: 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
№22 слайд
Содержание слайда: The time it takes a car to drive from Istanbul to Sinop is an example of a discrete random variable
True
False
№23 слайд
Содержание слайда: Probability is a numerical measure about the likelihood that an event will occur.
TRUE
FALSE
№24 слайд
Содержание слайда: 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
№25 слайд
Содержание слайда: If we flip a coin three times, the probability of getting three heads is 0.125.
TRUE
FALSE
№26 слайд
Содержание слайда: The number of products bought at a local store is an example of a discrete random variable.
TRUE
FALSE
№27 слайд
Содержание слайда: 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.
№28 слайд
Содержание слайда:
№29 слайд
Содержание слайда: Binomial Probability Distribution
Bi-nominal (from Latin) means: Two-names
№30 слайд
Содержание слайда: 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
№31 слайд
Содержание слайда: The Binomial Distribution
The binomial distribution is used to find the probability of a specific or cumulative number of successes in n trials
№32 слайд
Содержание слайда: The Binomial Distribution
The binomial formula is:
№33 слайд
Содержание слайда: Example:
Calculating a Binomial Probability
№34 слайд
Содержание слайда: 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)?
№35 слайд
Содержание слайда: 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
№36 слайд
Содержание слайда: 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
№37 слайд
Содержание слайда: 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
№38 слайд
Содержание слайда: Creating a probability distribution with the
Binomial Formula – house sale example
№39 слайд
Содержание слайда: Binomial Probability Distribution
house sale example
n = 5, P= .4
№40 слайд
Содержание слайда: The binomial distribution is used to find the probability of a specific or cumulative number of successes in n trials.
Let’s look at the cumulative probability: P (x < 2 houses), P(x 3)
№41 слайд
Содержание слайда: The binomial distribution is used to find the probability of a specific or cumulative number of successes in n trials.
Let’s look at the cumulative probability: P (x < 2 houses), P(x 3)
P ( x < 2 houses) = P(0 house) + P(1 house) = 0.0778 + 0.2592 = .337 or 33.7%
P(x 3 houses) = P(3 houses) + P(4 houses) + P(5 houses) = 0.2304 + 0.0768 + 0.0102 = 0.3174
№42 слайд
Содержание слайда: Shape of Binomial Distribution
The shape of the binomial distribution depends on the values of P and n
№43 слайд
Содержание слайда: 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.
№44 слайд
Содержание слайда: Using Binomial Tables instead of to
calculating Binomial probabilites
№45 слайд
Содержание слайда: 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)
№46 слайд
Содержание слайда: Solving Problems with Binomial Tables
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