10/28 Econ Midterm #2

Expected return a certain portfolio

E(ax+by) = a*E(x) + b*E(y)

Expected Value of x ̅ (unbiased)

E(x ̅ )=μ **where μ is the population mean

Expected value for binomial distribution

E(x) = μ = np

Expected Value for Uniform Probability Distribution

E(x)=(a+b)/2

Exponential Probability and the Poisson Distribution

The Poisson distribution (discrete) and the Exponential distribution (continuous) are closely related

variance for hypergeometric probability distribution

Var(x)=σ^2=n(r⁄N)(1-r⁄N)((N-n)⁄(N-1))

Probability DENSITY functions (continuous probability distribution)

a function of a continuous random variable, whose integral across an interval gives the probability that the value of the variable lies within the same interval

Variance of a discrete random variable: measure of variability

a weighted average of the squared differences between possible values for RV and the expected value, where weights are the associated probabilities • Var(x)=σ^2=∑(x-μ)^2 f(x) • Ex: x= # of heads after we flip a fair coin 3 times > Var(x) = (1.5)^2 * (1/8) + (1-1.5)^2 * (3/8) + (2-1.5)^2 * (3/8) + (3-1.5)^2 * (1/8) = 9/4 * 1/8 + 1/4 * 3/8 + 9/4 * 1/8 = 24/32 = 3/4 or 0.75

the probability density function (normal probability function)

an equation used to compute probabilities of continuous random variables

probability distribution of a discrete random variable

model for population data a) assign probabilities to an outcome or collection of outcomes b) usually written as a function > f(x) = probability

Conditions for Normal (continuous) Approx to a Binomial Distribution (discrete)

1) condition 1: np>= 5 2) condition 2: n(1-p) >= 5 Binomial ~ N(μ, σ)

Sampled population

the population from which the sample is drawn

Finite Population Correction Factor

used when you sample without replacement from more than 5% of a finite population. It's needed because under these circumstances, the Central Limit Theorem doesn't hold and the standard error of the estimate (e.g. the mean or proportion) will be too big.

when is diversification useful?

when correlation of returns is negative

correlation btwn random variables x and y

ρ_xy=σ_xy/(σ_x σ_y )

If the population is too small relative to sample, we need to add a finite population correction factor:

σ_x ̅ =√((N-n)/(N-1)) * (σ/√n) **where N is the population size

# of experimental outcomes providing exactly "x" successes in "n" trials

(nCx) = n!/ x!(n-x)!

Cumulative Distribution Function for the Exponential(Exponential Probability Distribution)

(the culminative probability of obtaining a value for the exponential rand var of less than or equal to some value denoted by x_0) P(x≤x_0 )=1-e^(-x_0⁄μ) **where x_0 is a fixed number. This gives us the probability of the random variable assuming any value below x_0. It's the same information as in the probability tables we've been using!

Standard deviation of two random variables (x and y)

(x-(E(x))(y-E(y))*p +.....

Poisson Probability Distribution formula

(μ^x e^-μ)/x! **Where e (Euler constant) is equal to 2.71828...; μ is the mean or expected number of occurrences in an interval; and x is the number of occurrences

Relationship between sample size and the sampling distribution

1) First: E(x ̅ )=μ regardless of the sample size, IF we're dealing with a random sample. 2) Second: the sample size affects the dispersion of the sampling distribution. The larger the sample size, the lower is the standard error.

EXAMPLES of CONTINOUS random variables

1) Rand Exp: Customer visits a web page> Rand Var: Time customer spends on web page per minutes > Possible Values for the Rand Var: x ≥ 0 2) Rand Exp: fill a soft drink can > Rand Var: # of ounces > Possible Values for the Rand Var: 0 ≤ x ≤ 12.1 3) Rand Exp: Test a new chemical process > Rand Var: Temp when the desired reaction takes place (min temp = 150; max temp = 212) > Possible Values for Rand Var: 150 ≤ x ≤ 212 4) Rand Exp: Invest $10K in the stock market > Rand Var: Value of investment after 1 yr> Possible Values for Rand Var: x ≥ 0

EXAMPLES of DISCRETE random variables

1) Rand Exp: flip a coin > Rand Var: face of coin showing > Possible Values for the Rand Var: 1 if heads; 0 if tails 2) Rand Exp: roll a die > Rand Var: # of dots showing on top of die > Possible Values for the Rand Var: 1,2,3,4,5,6 3) Rand Exp: Contact 5 customers > Rand Var: # of customers who place an order > Possible Values for Rand Var: 0,1,2,3,4,5 4) Rand Exp: Operate a health care clinic for one day > Rand Var: # of patients who arrive > Possible Values for Rand Var: 0, 1, 2, 3, .... 5) Rand Exp: Offer a customer the choice of two products > Rand Var: product chosen by customer > Possible Values for Rand Var: 0 if none; 1 if choose product A; 2 if choose product B

Uniform Probability Distribution EXAMPLE: Suppose x represents the travel time of a train from Chicago to Pittsburgh. The trip can last anywhere from 400 to 440 minutes. Suppose we have enough data to assert that any 1-minute interval has the same chance of occurring. 1. x is a continuous random variable; 2. x is said to follow a uniform probability distribution 3. the probability density function is: f(x)={(1⁄40, for 400≤x≤440; 0, for any other x

1) What is the probability that the train ride takes at least 405 minutes and at most 420 minutes? P(405<= x <=420) = 1/40 * (420-405) = 15/40 = 37.5% 2) What is the probability it takes at least 400 minutes and at most 440 minutes? P(400 <= x <= 400) = 1/40 * (440-400) = 40/40 = 1 3) What is the probability it takes exactly 425 minutes? P(x=425) = 1/40 * (425-425) = 0 **It's the same idea for every continuous probability distribution.

Properties of Normal Probability Distribution

1. Every single normal distribution is defined by two parameters: mean and standard deviation; 2. The mean is equal to the median, which is the highest point in the graph; 3. The mean could be any numerical value; 4. The distribution is symmetric (skewedness is zero). Moreover, x can be any value in the real line (the graph never touches the axis); 5. Standard deviation determines how flat and wide the normal curve is; 6. Probabilities are, as usual, determined by the area under the curve; 7. For the normal distribution: a) 68.3% of the values lie within 1 standard deviation of the mean b) 95.4% of the values lie within 2 standard deviations of the mean c) 99.7% of the values lie within 3 standard deviations of the mean

Properties of a Binomial Experiment

1. The experiment consists of n identical trials; 2. Two outcomes are possible on each trial (success or failure); 3. The probability of each outcome does not change from trial to trial (p is the constant probability of success); 4. The trials are independent.

Properties of a Poisson experiment

1. The probability of an occurrence is the same for any two intervals of same length (e.g. 30 minutes or 50 miles); 2. The occurrence in any interval is independent of the occurrence in any other interval.

Poisson Probability Example: You observe a call center during regular business hours for, say, 100 days. The average number of calls received in a 5-minute interval is equal to 12. If properties 1 and 2 are valid, the number of calls received during a 5-minute interval follows a Poisson distribution 2) What is the probability that we observe 4 calls in a 2.5-minute interval?

= 0.1338

Poisson Probability Example: Suppose the number of thunders on the coast of Scotland follows a Poisson distribution with an average of 10 occurrences every 60 minutes. What is the probability of observing exactly 4 thunders in any given half-hour?

= 0.1754 or 17.54%

Binomial Experiment Example: Suppose that Lehigh has 5,000 undergraduates, of which 1,000 have taken or are currently taking ECO45. From a list of emails, we randomly select (with replacement) a student and check if she/he has taken (or is taking) ECO45. We repeat this experiment 3 times. What is the chance that exactly 1 student is found to have had this amazing experience?

= 0.384

normal distribution (continuous probability distribution)

A function that represents the distribution of variables as a symmetrical bell-shaped graph.

Random sample (infinite population)

A simple random sample of size n from an infinite population is a sample selected such that the following conditions are satisfied: 1. Each element selected comes from the same population. 2. Each element is selected independently. **Ex: how would you select a random sample of customers arriving at a restaurant. Is this a finite population? How to ensure that each element is selected independently?

continuity correction factor

A value of .5 that is added to or subtracted from a value of x when the continuous normal distribution is used to approximate the discrete binomial distribution.

Expected value for hypergeometric probability distribution

E(x)=μ=n(r⁄N)

Standard Normal Probability Distribution Examples 1 + 2

Example 1: let z be a random variable with a standard normal probability distribution. What is the probability that z assumes the value 0.5 or lower? P(E <= 0.5) = 0.69146 or 69.14% Example 2: What is the probability that z assumes the value -0.5 or lower? P(E<= -0.5) = P(E>=0.5) = 1-P(E<=0.5) = 1-0.69146 = 0.31 or 31%

Probability functions (continuous probability distribution)

Functions that assign probabilities to values of a random variable, determine the information in a probability distribution. Needs to have the sum of the probability values equal to one and also needs each probability value to be greater than zero and less than one.

Population does have a Normal Distribution

In this case, the sampling distribution of x ̅ is normally distributed for any sample size

normal probability table

Lists z-scores and corresponding percentiles

How to go from any normal to a standard normal distribution(prob)

N(a, b) > z = x-μ / σ - z_a= a-a/b -z_b = (a+b)-a / b

Hypergeometric probability distribution example: In a classroom of 30 students, 10 of them scored 90 or higher in a final exam. The instructor forms a group of three students by randomly selecting, without replacement, among the class. What is the probability that exactly one student in this randomly selected group scored 90 or higher?

N= 30 n =3 r = 10 f(1) = (10,1) (30-10, 3-1) / (30,3) ~ .4679

Binomial Probability Distribution Formula

P(x)= (nCx) (p^x) (1-p)^n-x **Where x is the number of successes; n is the number of trials; p is the probability of success in any given trial

Exponential Probability and the Poisson Distribution Example 1: Suppose that the number of potholes in any 4-mile stretch of a highway follows a Poisson distribution with mean 10. Hence:

Poisson f(x) = (μ^x * e^-μ)/x! = 10^x*e^-10 /x!

Types of Discrete Probability Distributions

Uniform, binomial, Poisson, hypergeometric

Standard Deviation a certain portfolio (or variance of a linear combo of 2 Rand Vars x and y)

Var(ax + by) = a^2Var(x) + b^2Var(y) +2ab*σ_xy ==> then sqrt answer

Variance for Uniform Probability Distribution

Var(x)=〖(b-a)〗^2/12

uniform probability distribution

a continuous probability distribution for which the probability that the random variable will assume a value in any interval is the same for each interval of equal length

Exponential Probability Distribution

a continuous probability distribution that describes the interval (of time, length) between occurrences

normal probability distribution

a continuous probability distribution. Its probability density function is bell-shaped and determined by its mean and standard deviation .

standard normal distribution

a normal distribution with a mean of 0 and a standard deviation of 1

Random variables (discrete probability distributions)

a numerical description of the outcomes of an experiment 1) case 1: numbers are neutral > Ex: roll a dice twice. Possible outcomes: 36; Random variable (x): sum of the faces; possible outcomes: (2 to 12) 2) case 2: we can assign numbers ourselves • Ex: student assessment (good = 0, extra good = 1, excellent = 2, super excellent =3)

discrete uniform probability distribution

a probability distribution for which each possible value of the random variable has the SAME PROBABILITY • f(x) = 1/n, where n - # of values r.v. may assume

Bivariate Distributions

a probability distribution involving two random variables • Ex: sum of numbers in two die rolls and result of a (fair) coin flip (0 is T, 1 is H) • Ex: number of rainy days in a year and yearly return of a commodity index

Poisson Probability Distribution

a probability distribution showing the probability of x occurrences of an event over a specified interval of time or space

Binomial Probability Distribution

a probability distribution showing the probability of x successes in n trials of a binomial experiment

DISCRETE random variable (discrete probability distributions)

a random variable that may assume either a FINITE number of values or an INFINITE SEQUENCE of values - (-3, -2, 1, 7, 10); (1,2,3,4,5,....) • Ex: # of classes attended by a randomly selected student • Ex: # of galaxies in the universe

Simple random sample (finite population)

a simple random sample of size n from a finite population of size N is a sample selected such that each possible sample of size n has the same probability of being selected **in practice, not so easy. Example: mammography trials in the 1960s.

Poisson probability table

a table that we can use under certain conditions that will make calculating probabilities a little easier when using the Poisson Distribution.

Expected value (or mean) for a random variable: measure of central location

a weighted average of all possible values for the RV, where weights are the associated probabilities • E(x) = μ = Σxf(x) • Ex: x= # of heads after we flip a fair coin 3 times > E(x) = 0*(1/8) +1*(3/8) + 2*(3/8)+3*(1/8) = (3+6+3)/8 = 12/8 = 1.5

Poisson Distribution EXAMPLE: Consider a Poisson distribution with mean equal to 3. a. Write the appropriate Poisson probability function. Recall that f(x)=(μ^x e^(-μ))/x! b. Compute f(2). c. Compute f(1). d. Compute P(x>=2)

a) f(x) = (3^x e^-3)/x! > f(0) = 3^0 e^-3/0! = 0.0497 b) f(2) = 3^2 e^-3 /2! = 0.224 or 22.4% c) f(1) = 3^4 e^-3/1! = 0.1493 or 14.9% d) P(x>=2) = 1-(P(x<2) = 1-P(x=0 U x=1) > 1-(P(x=1) + P(x=0)) = 1- (0.1493 +0.0487) = 0.801 or 80%

CONTINUOUS random variable (discrete probability distributions)

could be any value in an INTERVAL • measurements: temp, weight, height, ... P([108,109]) • Ex: weight of a tuna in a Tokyo market (108.743212.. kg)

normal probability density function (standard normal probability distribution)

describes a symmetric, bell shaped curve; completely defined by the mean and variance (standard deviation)

Normal Approximation to a Binomial Distribution Example: the internal audit team of a large company knows that there is, on average, a 20% chance that any client interaction will be non-compliant with a particular corporate policy. Suppose the audit team decided to audit a random sample of 100 such interactions. What is the chance that exactly 15 of the audited interactions will be labeled non-compliant?

f(15) = (100C15) * (0.2)^15 * (0.8)^85 = 0.04806 or 4.81%

Exponential Probability Distribution > a continuous probability distribution that describes the interval (of time, length) between occurrences - Example: suppose that x represents the loading time for a truck in a dock (or the interval of time between two occurrences, loading a truck). Suppose further that x follows an exponential distribution with an expected value of 15 minutes.

f(x) = 1/μ * e^-x/μ = 1/15 * e^-x/15

Poisson Probability Example: You observe a call center during regular business hours for, say, 100 days. The average number of calls received in a 5-minute interval is equal to 12. If properties 1 and 2 are valid, the number of calls received during a 5-minute interval follows a Poisson distribution 1) What is the probability that we observe 8 calls in a 5-minute interval?

f(x)=(12^x e^(-12))/x!

Hypergeometric probability distribution formula

f(x)=(r¦x)((N-r)¦(n-x))/((N¦n) ) **Where x is the number of successes (having scored 90+); n is the number of trials (3 selected students); N is the number of elements in the population (30 students); and r is the number of successes in the population (10 students are known to have scored 90+).

Exponential Probability Distribution formula (It's a continuous probability distribution that describes the interval (of time, length) between occurrences)

f(x)=1/μ e^(-x⁄μ) **where μ is the expected value and is x≥0 is the random variable (e.g., length of time between occurrences)

uniform probability density distribution formula

f(x)={(1/(b-a), for a≤x≤b, 0, for other x)

selection bias

in an experiment, unintended differences between the participants in different groups

Population does not have a Normal Distribution

in these cases, we can invoke the CENTRAL LIMIT THEOREM: for random samples of size n from a population, the sampling distribution of the sample mean x ̅ can be approximated by a normal distribution as the sample size becomes large

Population has a Normal Distribution

in this case, the sampling distribution of x ̅ is normally distributed for any sample size

Frame

list of elements from which the sample will be selected from **we could sample from a finite or an infinite population

Mean for Normal Approximation to a Binomial Distribution

mean = n*p = μ

bivariate empirical discrete probability/ bivariate probabilities distribution or joint probabilities

represents the joint probability distribution of a pair of random variables > Each row in the table represents a value of one of the random variables (call it X) and each column represents a value of the other random variable (call it Y)

Standard Deviation for Normal Approximation to a Binomial Distribution

sqrt(n*p(1-p)) = σ

Variance of a Random Variable

square each value and multiply by its probability

Normal Approximation to a Binomial Distribution

suppose you're studying a binomial experiment. Whenever np≥5 and n(1-p)≥5, then the normal distribution is a good continuous approximation to the discrete binomial distribution

expected value

the average of each possible outcome of a future event, weighted by its probability of occurring

Central limit theorem

the distribution of sample averages tends to be normal regardless of the shape of the process distribution > as the size n of a simple random sample increases, the shape of the sampling distribution of x̄ tends toward being normally distributed

Normal Distributions with same means, but different standard deviations

the larger the standard deviation, the more dispersed, or spread out, the distribution is

hypergeometric probability distribution

the probability distribution that is applied to determine the probability of x successes in n trials when the trials are not independent 1. The trials are not independent; 2. Probability of success changes from trial to trial. • Ex: In a classroom of 30 students, 10 of them scored 90 or higher in a final exam. The instructor forms a group of three students by randomly selecting, without replacement, among the class. What is the probability that exactly one student in this randomly selected group scored 90 or higher?

Standard deviation of a random variable

the square root of the variance

Normal Distributions with the same SD, but different means

these curves have the same shapes but are located at different positions on the x axis

Binomial Probability Table

used to calculate probabilities instead of using the binomial distribution formula. The number of trials (n) is given in the first column. The number of successful events (x) is given in the second column.

Variance for Poisson probability table

σ^2=μ

Standard Deviation of x ̅

σ_x ̅ =σ/√n **where is the population standard deviation (σ) and is the sample size (n)

Sampling Distribution of x ̅ Example: In the managers example, the population standard deviation was equal to 3999.2 and the sample size was 30. Hence, the standard error of the mean is equal to:

σ_x ̅ =σ/√n=3999.2/√30≈730.15

covariance of random variables x and y

σ_xy=[Var(x+y)-Var(x)-Var(y)]⁄2 or σ_xy=∑_(i,j)[x_i-E(x_i )][y_j-E(y_j )]f(x_i,y_j )

variance for binomial distribution

σ² = np(1-p)

Sampling Distribution

• Every time we draw a random sample, we get different point estimators. If we consider taking a sample to be an experiment, then the point estimators are random variables. • As such, the point estimators have an expected value and variance, and can have an associated probability distribution. This is the ALL-IMPORTANT sampling distribution

Sampling Distribution of x ̅

• x ̅ is a random variable, since any one sample will have a (somewhat) different mean. • The probability distribution of x ̅ is called the sampling distribution. • The sampling distribution of x ̅ is the probability distribution of all possible values of the sample mean x ̅