Data analysis: Chapter 6: Discrete Probability Distributions

The average number of customers at Jimmy's Burgers is 1.7 per minute. What is the probability that only 1 customer arrives in the next minute?

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Discrete random variables

1. are countable 2. can have a finite or infinite number of values 3. have a set of distinct values

probability distribution function (PDF) / cumulative distribution function (CDF)

defined either by a list of X-values and their probabilities or by mathematical equations - a discrete PDF shows the probability of each X-value, while the CDF shows the cumulative sum of probabilities, adding from the smallest to the largest X-value

binomial shape

skewed right if π < .50, skewed left if π > .50, and symmetric only if π = .50.

In the formula to calculate a probability of the hypergeometric distribution, what does the denominator represent?

the number of ways a sample size of n can be selected from the population of size N

standard deviation

the square root of the variance is is denoted by σ

expected value of a discrete random variable

the sum of all X-values weighted by their respective probabilities

Which of the following are examples of discrete random variables?

1. the number of people who vote for the democratic candidate in the next presidential election 2. the number of students who received an "A" on their statistics exam.

XYZ Corp. makes widgets. 1% of the widgets are defective. XYZ manufactures 100,000 widget, the number of defective widgets is expected to be __________.

100,000*0.01=1000

XYZ Corp. makes widgets. 1% of the widgets are defective. XYZ manufactures 100,000 widgets. The standard deviation for this binomial distribution is _________.

square root of 100,000*.01*.99 = square root of 990 = 31.46

In which of the following situations is it appropriate to use a Poisson process?

the number of calls arriving to a customer help line in a one hour period

True or false: σ2 represents the variance

true

A company receives an average of 0.8 purchase orders per minute. The company wants to determine the probability of receiving 6 purchase orders in 5 minutes. What value should the company use for the mean to help calculate the probability?

.8*5=4

The probability of any x value of a discrete distribution is _____

0≤P(xi)≤1

Parameters

random variables and their distributions

geometric distribution

related to the binomial - describes the number of Bernoulli trials until the first success is observed. - always skewed to the right - geometric probabilities sum to 1

Bernoulli experiment

a random experiment that has only two outcomes - Named after Jakob Bernoulli (1654-1705)

True or False: E(X) = μ

True

Which of the following random variables meets the criteria for a hypergeometric distribution?

Out of 50 adults, 10 who have a graduate degree. A sample of 20 is taken. Define X to be the number of adults in the sample with a graduate degree.

The average number of customer arriving at Jimmy's Burgers in a minute is 1.7. Which expression would one use to calculate the probability that at east 4 customers arrive in a randomly chosen minute?

1-P(X< or equal to 3)

In the binomial distribution which expression represents the probability of failure?

1-π

Which of the following should one look for when identifying a hypergeometric application?

1. A finite population, N 2. A known number of success, s

Which of the following is an example of a binomial experiment?

1. Ask customers at a movie theater if they spent $20 of more on concessions 2. Asking randomly selected people if they are a member of facebook

parameters of the hypergeometric distribution

1. N= number of items in the population 2. n= sample size 3. s=number of successes in population

4 main characteristics of Poisson Applications

1. an event of interest occurs randomly over time or space 2. the average arrival rate (λ) remains constant 3. the arrivals are independent of each other 4. the random variable (X) is the number of events within an observed time interval

Which of the following are characteristics of a binomial distribution?

1. each trial is independent of the previous trial 2. for each trial the probability of success remains the same 3. each trial has only two possible outcomes

The symbol π in the binomial PDF

1. is in the interval (0,1) 2. is the probability of success

Generally, a hypergeometric distribution will be used instead of a binomial distribution when:

1. sampling is done without replacement 2. there is a finite population 3. independence of the trials is uncertain

In which of the following ways are binomial distributions and hypergeometric distributions similar?

1. the both have two possible outcomes: success or failure 2. they are both discrete distributions

Which of the following are characteristics of a discrete uniform distribution?

1. the distribution is symmetric 2. the probability of each value of the random variable is the same 3. the random variable has a finite number of outcomes

In which of the following situations is it appropriate to use a Poisson process?

1. the number of customers who purchase concessions every 5 minutes, while a movie is playing at a theater 2. the number of landfills per county in the state of Texas

Which of the following are characteristics of a Poisson distribution?

1. the number of occurrences must be an integer 2. each occurrence is independent of other occurrences 3. the intervals that the occurrences are counted in should be the same length

Poisson distribution

describes the number of occurrences within a randomly chosen unit of time - the events must occur randomly and independently over a continuum of time or space - The mean of the Poisson distribution is λ, and its standard deviation is the square root of the mean - All characteristics of the Poisson model are determined by its mean λ

A bimodal distribution is skewed left when the probability of success is ________.

greater than .5

discrete random variable

has a countable number of distinct values

random variable

is a function or rule that assigns a numerical value to each outcome in the sample space of a random experiment

recognizing hypergeometric applications

look for a finite population (N) containing a known number of successes (s) and sampling without replacement (n items in the sample0 where the probability of success is not constant for each sample item drawn

A list of the top twenty restaurants in Chicago was released. Four of the restaurants specialize in seafood. If five of the restaurants are selected randomly from the list, the expected value for the number of restaurants specializing in seafood is _______.

n(s/N) = 5(4/20) = 1

Which of the following criteria indicate it would be acceptable to approximate the hypergeometric with the binomial?

n/N < .05

variance of a discrete random variable

the sum of the squared deviations about its expected value, weighted by the probability of each X-value. - is a weighted average that measures variability about the mean

Random variables are represented by _____-case letters, while particular values of random variables are represented by ___________-case letters.

upper; lower

An apartment complex rents an average of 2.3 new units per week. If the number of apartments rented each week is Poisson distributed, then the probability of renting no more than 1 apartment in a week is ________.

0.331

The sum of the probabilities of all x values in a discrete distribution equals ______.

1

5 main characteristics of a binomial situation

1. the number of trials (n) is fixed 2. there are only two outcome for each trial: success or failure 3. the probability of success for each trial remains constant 4. the trials are independent of each other 5. the random variable (X) is the number of successes out of n trials

Which of the following experiments is likely to produce a uniform discrete distribution?

1. the values that occur from repeated spins of a roulette wheel at a casino 2. the answer selected on a multiple choice question that has four choices by a student who did not study

The binomial distribution can be approximated using the Poisson ________

1. λ=nπ 2. when n is large and π is less than or equal to .05

20% of a restaurant's customers order the chef's special. 230 customers are anticipated to dine tonight at the restaurant. The expected number of chef's specials that will be ordered tonight is _______.

230*0.2=46

True or false: A Bernoulli experiment has three possible outcomes

False

20% of a restaurants customers order the chef's special. 230 customers are anticipated to dine at the restaurant tonight. The standard deviation for this binomial distribution is _______ (round you answer to 3 decimal places.)

SqRt of 230*0.2*0.8=6.066

binomial distribution

arises when a Bernoulli experiment is repeated n times. Each Bernoulli trial is independent so that the probability of success π remains constant on each trial

probability model

assigns a probability to each outcome in the sample space defined by a random process - used to depict the essential characteristics of a stochastic process, to guide decisions or make predictions

discrete probability distribution

assigns a probability to each value of a discrete random variable X. - follows the rules of functions

A _____ random variable is the sum of repeated Bernoulli trials.

binomial

uniform distribution

one of the simplest discrete models - describes a random variable with a finite number of consecutive integer values from a to b. - the entire distribution depends only on the two parameters a and b. - each value is equally likely

A Poisson process is used to observe the number of occurrences ______.

over time or in space

The acronym PDF means ________ distribution function and the acronym for CDF means ______ distribution function.

probability; cumulative

hypergeometric distribution

similar to the binomial except that sampling is without replacement from a finite population of N items. Therefore, the trials are no independent and the probability of success is not constant from trial to trial

A company receives an average of .64 purchase orders per minute. Assuming a Poisson distribution for the number of purchase orders per minute, what is the standard deviation for the distribution?

square root of .64 = .8