Ch. 6

Criteria for Poisson Application (4)

1. Event occurs randomly over time and space 2. Average arrival rate (𝝺) remains constant 3. Arrivals are independent of each other 4. The random variable (X) is the number of events within an observed time interval.

Example: You flip a coin 10 times and count the number of heads (X) - Does X have a binomial distribution?

1. Fixed number of trials? Yes - flipping 10 times | n = 10 2. Does each trial have only two possible outcomes (success | failure)? Yes = heads | tails 3. Is P (probability of success) the same for each trial? Yes = coin is fair and not weighted 4. Are the trials independent? Yes - you assume that coin is being flipped the same way in each trial, so the outcome of one flip doesn't affect the outcome of subsequent flips. You flip a coin 10 times and count the number of heads (X) - Does X have a binomial distribution? Answer: Because the random variable X that occurs in 10 trials meets all 4 conditions, it has a binomial distribution with n = 10 and P = 1/2

Characteristics of a Probability Distribution (3):

1. The probability of a particular outcome is between 0 and 1 inclusive. 2. The outcomes are mutually exclusive (Can't happen at the same time). 3. The list of outcomes are exhaustive. - So, the sum of the probabilities of the outcomes is equal to 1.

4 Steps to Identify whether a random variable is binomial:

1. There are a fixed # of trials 2. Each trial has 2 possible outcomes (success | failure) 3. Probability for success is the same for each trial 4. Trials are independent (outcome of one trial doesn't influence that of any other)

A medical laboratory receives 34 blood specimens to check for HIV. Nine actually contain HIV. A worker is accidentally exposed to five specimens. What is the probability that fewer than 3 contained HIV?

P(X < 3) = P(X ≤ 2) = HYPGEOM.DIST(2,5,9,34,1) = .8977.

A medical laboratory receives 34 blood specimens to check for HIV. Nine actually contain HIV. A worker is accidentally exposed to five specimens. What is the probability that none contained HIV?

P(X = 0) = HYPGEOM.DIST(0,5,9,34,0) = .1909.

The probability that an American CEO can transact business in a foreign language is .20. Ten American CEOs are chosen at random. What is the probability that all 10 can transact business in a foreign language?

P(X = 10) = BINOM.DIST(10,10,.20,0) = .00000

Keep an eye out for wording: Fewer than 2 times =

P(X ≤ 1) - at most "1"

Calculate the probability of at least 1 successes in 9 trials with a 30 percent chance of success:

P(X ≥ 1) = 1 - P(X ≤ 0) = BINOM.DIST(x,n,π, 1) P(X ≥ 1) = 1 - P(X ≤ 0) = 1 - BINOM.DIST(0,9,0.30,1) = 0.9596.

The probability that an American CEO can transact business in a foreign language is .20. Ten American CEOs are chosen at random. What is the probability that at least two can transact business in a foreign language?

P(X ≥ 2) = 1 - P(X ≤ 1). Using Excel: = 1 - BINOM.DIST(1,10,.20,1) = .6242.

If you can split the measurement in half and it still makes sense, then the variable is...

continuous: - ½ a minute = 30 seconds - ½ a foot = 6 inches

- If the possible outcomes of a random variable can be listed out using whole numbers (0, 1, 2, 3, 4) or (1, 2, ...) - Then the random variable is...

discrete

If you split it and it doesn't make sense, then the variable is...

discrete: - Half a person? - Half a role (half CEO)? - Half an accident?

- Each Bernoulli trial is...

independent

Finite -

list of possible values has a fixed (finite) number of elements in it - Ex: The number of smoking ban supporters in a random sample of 100 voters - has to be between 0 and 100: (0, 1, 2, ...100)

Probability Distribution Can be described by measures of location and dispersion - By computing the distribution's...

mean, variance, and standard deviation.

- In a binomial experiment, we are interested in X = # of successes in n trials - So the binomial random variable X is the sum of _ independent random variables.

n

- Binomial distribution arises when a Bernoulli experiment is repeated n times.

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- Parameter = λ (Lambda), it Represents...

the mean number of events per unit of time or space

- Each Bernoulli trial is independent, this means that...

the probabilities must remain the same throughout the trials; each event must be completely separate and have nothing to do with the previous event.

Bernoulli Experiment Continued... - Probability for success is ___ _____ for each trial. - The only parameter needed to define a Bernoulli process is 𝜋 - It's an important building block in distribution models. - Success doesn't mean success in the usual way — it just refers to an outcome you want to keep track of. - For example, you might want to find out how many boys are born each day, so you call a boy birth a "success" and a girl birth a "failure." - In the dice rolling example, a double six die roll would be your "success" and everything else rolled would be considered a "failure." ** Not the same as the Bernoulli Principle

the same

- Parameter =

λ (Lambda)

Clues to recognize a PDF or a CDF?

(PDF) Probability Distribution Function ...shows the probability of each X-value.(CDF) Cumulative Distribution Function ....shows the cumulative sum of probabilities... PDF (Each) Probability that none of... = P(X=0) Probability that two of... = P(X=2) Probability that sample of 6... = P(X=6) CDF (Cumulative) Probability that fewer than 2... = P(X < 2) Probability that at least 4... = P(X ≥ 4) Probability that at most 8... = P(X ≤ 8)

A medical laboratory receives 34 blood specimens to check for HIV. Nine actually contain HIV. A worker is accidentally exposed to five specimens. What is the probability that at least 2 contained HIV?

(X ≥ 2) = 1 - P(X ≤ 1) = 1 - HYPGEOM.DIST(1, 5,9,34,1) = .3999.

Poisson Distribution and Time - "Time", the most common application is... - Let X = - Value of X is a... - Often called "model of arrivals"(customers, defects, accidents) - Each event is...

- "Time", the most common application is modelling arrivals per unit of time. - Let X = the number of events per unit of time. - Value of X is a random variable that depends on when the next unit of time is observed. - Often called "model of arrivals" (customers, defects, accidents) - Each event is independent

Probability Distribution-

- A listing of all the outcomes of an experiment and the probability associated with each outcome. - Like relative frequency distribution; however, instead of describing the past, it's used to provide estimates of the likelihood of future events.

What is a PDF or CDF?

- A probability distribution function (PDF) is a mathematical function that shows the probability of each X-value. - A cumulative distribution function (CDF) is a mathematical function that shows the cumulative sum of probabilities, adding from the smallest to the largest X-value, gradually approaching unity.

Binomial Distribution - Bernoulli Experiment:

- A random experiment that has only 2 outcomes - To create a random variable, we arbitrarily call - one outcome a "success" (X=1) - the other outcome a "failure" (X=0) Probability of success - Denoted in 𝜋 (Greek letter "pi", not 3.14159) - X = 1 | 𝜋 Probability of failure - 1 - 𝜋 - Probabilities must sum to 1 = P(1) = (1 - 𝜋) + 𝜋 = 1

Mean of a random variable-

- Average of all the outcomes you would expect in the long run (over all possible samples - X)

Uniform Distribution:

- Data set is evenly distributed among the values. - Does NOT mean that all the values are the same. - Means frequencies for the data are the same across the groups

Poisson Distribution-

- How likely is it that out of the ordinary things will happen? - Describes the number of occurrences within a randomly chosen unit of time or space.

- 3 Parameters for a Hypergeometric Distribution:

- N = number of items in the population. - n = number of items in the sample - s = number of successes in the population

Variance of a Random Variable X:

- Roughly means - average squared distance from the mean for all the outcomes you would get in the long run, over all possible values. - Same as variance of the population of all possible values. - Notation: - σ² - Sigma squared or - σ²ₓ - Sigma sub x squared

Hypergeometric Distribution:

- Similar to binomial distribution except that sampling is without replacement from a finite population of N items. - Trials are NOT independent - Probability of success is NOT constant

Binomial Shape: - Skewed right if - Skewed left if - Symmetric if

- Skewed right if 𝜋 <.50 - Skewed left if 𝜋 >.50 - Symmetric if 𝜋 = .50

Standard Deviation of Random Variable X:

- Standard deviation shows the variation in the data - Data close together = standard deviation is small - Data far apart = standard deviation is large - Square root of the variance - Roughly represents the average distance from the mean - Notation: σ = √σ² = √var(x)

Expected Value (X) of a discrete random variable is...

- Sum of all the X-values - Weighted by their respective probabilities - It's a measure of center

Continuous Random Variables:

- Typically represent measurements - Ex: Time to complete a task | Weight of a newborn - They are different from discrete random variables because they are uncountably infinite. - Too many possible values to list out and/or - Can be measured to a high level of precision (level of smog in LA on a given day)

- Notations: - X = - x =

- X = random variable in S - x = specific values of X (x₁, x₂)

Discrete Uniform Distribution is used to...

- generate random integers that are then used to randomly sample from a population. - Yay Excel! =RANDBETWEEN(a,b) - Example: Generate a random number between 5 and 10 - =RANDBETWEEN(5,10) Tip: Generate a few extras since the same integer may come up more than once. - Useful in audits - this allows an auditor to choose numbered invoices at random.

A medical laboratory receives 34 blood specimens to check for HIV. Nine actually contain HIV. A worker is accidentally exposed to five specimens.

= HYPGEOM.DIST(x,n,s,N,0) x=tbd | n=5 | s=34 | PDF = 0

Calculate the Poisson probability: Fewer than 10 arrivals with λ = 6.40

= POISSON.DIST (x,λ,1) where "1" stands for cumulative P(X < 10) = P(X ≤ 9) = POISSON.DIST (9, 6.40,1) = 0.88580.

Adjusting for Probability of "Fewer Than": "Fewer than 3" =

=P(X<3) Actually means...less than or equal to 2 = P(X<2) - Can't be 3...but can be 2 or less ...So P(X<3) = P(X <_ 2)

Random Variable -

A variable measured or observed as a result of an experiment.

- 3 frequently occurring discrete probabilities:

Binomial, hypergeometric, Poisson

Expected Value Application: Raffle Tickets - The cost of one raffle ticket is $2 - The prize is a luxury automobile worth $55,000 - 29,346 tickets are sold - What's the expected value of the lottery ticket?

E(X) = (value if you win)P(win) + (value if you lose)P(lose) = (55,000)(1/29,346) + (0)(29,345/29,346) *because 1 winner = (55,000)(.000034076) + (0)(.999965924) = $1.87

- Criteria for a Poisson distribution:

Events must occur randomly and independently over a continuum of time or space.

2 Classes of Discrete Random Variables:

Finite and Countably Infinite

A statistics textbook chapter contains 65 exercises, 11 of which are essay questions. A student is assigned 15 problems. What is the probability that two or more are essay?

Hint: two or more is the same thing as "at least 2" = HYPGEOM.DIST(x,n,s,N,1) P(X ≥ 2) = 1 - P(X ≤ 1) = 1 − HYPGEOM.DIST(1,15,11,65,1) = .7861.

Expected Value =

Mean of Probability Distribution

Special Note for Calculating Probability of at least something... ">_" Example: Probability that there will be at least 3 successes = P > 3 Changes to:

P >_ 3 = 1 - (P <_ 2)

There is a 79 percent chance that an airline passenger will check bags. In the next 20 passengers that check in for their flight at Denver International Airport, find the probabilities that AT LEAST 7 will check bags. n = 20 and π = 0.79 Find the X...

P(X ≥ 7) = 1- P(X ≤ 6) So x = 6 | n = 20 | π = 0.79 | CDF so last digit is 1 P(X >_ 7) = 1 - P(X <_ 6) = 1- BINOM.DIST(x,n,π, 1) P(X ≥ 7) = 1- P(X ≤ 6) = 1- BINOM.DIST(6,20,0.79, 1) = 1.0000.

Probability of a specific outcome=

P(x) => P of x = signifies probability that the random variable X takes on a specific value which is the little "x" = probability of getting 1 on a die is P(x), P(1)

To be a valid probability distribution, the following 2 things must be satisfied:

The probability of a particular outcome is between 0 and 1 inclusive. The list of outcomes are exhaustive. So, the sum of the probabilities of the outcomes is equal to 1.

Special Note for Calculating Probability of at least something... ">_" Example: Probability that there will be at least 3 successes = P >_ 3 Changes to: P >_ 3 = 1 - (P <_ 2) Why?

This is a CDF (Cumulative) probability calculation. CDFs always calculate probabilities that are "less than or equal to" or "<_" to something. So you have to change the probability to account for that. So...Greater than or equal to 3, or At least 3, or >_ 3 is rewritten as... P >_ 3 = 1 - (P <_ 2)

If 5 percent of automobiles in San Carlos have one burned-out headlight, what is the probability that, in a sample of 10 automobiles, none will have a burned-out headlight?

X = burned out headlight n = 10 and π = .05 find P(X = 0) Yay Excel! =BINOM.DIST(x,n,π,0) =BINOM.DIST(0,10,.05,0) = .59874.

Uniform Distribution is one of the simplest discrete probability 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 _ and _ - Each value is equally _______

a and b, likely

Discrete Probability Distribution -

assigns a probability to each value of a discrete random variable X

Discrete Random Variable X - Can take on a certain set of possible outcomes - Each of these outcomes has a certain probability of occurring (x) Example: Rolling a Die - Look at the outcome - Random variable X = outcome of the die - X takes on the possible values of (1, 2,...6) - If you roll the die and you get a "1" = specific outcome = x₁

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If using Excel, you can use the formulas: - =BINOM.DIST(x,n,π,0) - for PDfs - =BINOM.DIST(x,n,π,1) - for CDFs

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Let X equal the total # of successes in n trials. - If all 4 criteria are met, X has a binomial distribution with probability of success on each trial equal to P

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Other practical uses for knowing how to compute the expected value. - Actuaries - insurance people who compute premium rates, dividends, risks, etc., according to probabilities based on statistical records - Actuarially fair insurance program must collect as much in overall revenue as it pays out in claims. - Set the premiums to reflect empirical experience with the insured group. Individuals may gain or lose, but if the pool of insured persons is large enough, the total payout is predictable.

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The binomial formula is cumbersome even when you have a small n. Preferred method is Excel, but when you don't have access to Excel, you can use Appendix A to look up binomial probabilities for selected values of n and 𝜋.

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Why do we care? - Need to learn how probability can be used to analyze business actions and activities that generate random data. - Probability distributions can be a great tool for estimating future returns and profitability. - Probability distributions can be used to create scenario analyses. A scenario analysis uses probability distributions to create several, theoretically distinct possibilities for the outcome of a particular course of action or future event: Best-case, Likely, Worst-case - Also a great tool for risk analysis.

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Countably Infinite -

possible values can be listed out, but they have no specific end (goes on and on) - Ex: # of car accidents in a 10 year period - You know there is a largest value somewhere, can't say where exactly, so you list them all (1001, 1002,...)

____ assigns a numerical value to each outcome in the samples space of a random experiment.

rule

Remember: Variance =

square root, - Average squared deviation from the mean of a set of data - Can't be easily interpreted - So you use standard deviation for interpretation - Because it's in the original units of X

Calculate the Poisson Probability: At least 9 arrivals with λ = 5.40 = POISSON.DIST (x,λ,1) where "1" stands for cumulative Find the x (you have everything else!)

λ = 5.40, P(X ≥ 9) = 1 - P(X ≤ 8)....so x = 8. Final answer:P(X ≥ 9) = 1 - P(X ≤ 8) =1 − POISSON.DIST( 8, 5.40,1) = 0.09735.

Notation for the Mean of a random variable:

μ (mu sub x | μₓ) - Since you are looking at all the outcomes in the long run, it's the same as looking at the mean of entire population of values - Why you denote it as μ or μₓ - You put the x in the subscript to remind you that the variable this mean belongs to is the X variable.

- Because Expected value is an average/mean, we use the _ symbol, E(X)= _

μ,μ