Class 08 - 2016-10-31

General

• Two characteristics to data

1. Measure of Central Tendency
2. Measure of dispersion with respect to 1 above.

• Two types of error can arise:

1. Bias: error from sampling nonrandomly
2. Error: error from the chance that a sample (randomly selected) is not representative of the population

• Central Limit Theorem: If the sample size is sufficiently large (over 30) then the mean of the x-bar has a sampling distribution that is approximately bell shaped, regardless of the shape of the distribution.
• Standard error of the sampling distribution of x-bar is sigma/sqrt(n)

Chapter 5

• Discrete distribution => values of x are countable

5.1: Probability Distribution for a discrete variable

• A mutually exclusive list of all the possible numerical outcomes along with the probability occurrence of each outcome
• Expected Value of a discrete Variable: The measure of central tendency
• Variance of a discrete variable (lowercase sigma squared): multiply each possible squared difference by its corresponding probability
  • std deviation of a discrete variable is the sqrt of variance

5.2: Covariance of a Probability Distribution and its application in Finance

• Covariance of a probability distribution measures the strength of the relationship between 2 variables
• Do 2 stocks move together or not? Inverse relationship means diversified portfolio.

5.3 Binomial distribution

• pass

5.4 Poisson Distribution

• Area of opportunity - the time or space interval
• Characteristic lambda = mean or expected number of events per unit
  • variance also = lambda, std dev
• Examples:
  • surface defects on a new refrigerator
  • number of network failures in a day
  • number of people arriving at a bank
  • number of fleas on a dog
• Use poisson distrib if these 4 cases hold:
  • You are interested in counting the number of times a particular event occurs in a given area of opportunity (defined by time, length, surface area, etc)
  • Equal probability distribution across all area of opportunity
  • events are independent
  • probability that two or more events will occur in an area of opportunity approaches zero as the area of opportunity gets smaller.

5.5 Hyper geometric

• pass

Chapter 6: Normal Distribution and other distributions

• Continuous distribution: values of x are not countable, but rather measurable
• standard normal is z-score-ified
• NORMSINV is the inverse of the CDF of the standard normal distribution
  • scipy.stats.norm.ppf object
    • Uses mean=0 and stddev=1, which is the "standard" normal distribution.
    • Use a different mean and standard deviation by specifying the loc and scale arguments
    • scipy.stats.norm.cdf is inverse
    • The acronym ppf stands for percent point function, which is another name for the quantile function.

6.2 Normal Distribution

• Total interquartile range is 4/3 standard deviations
• Middle 50% contained within mu +/- 2/3 sigma
• The range is equal to 6 standard deviations

6.3: Evaluating Normality

• To determine whether a set of data can be approximated by the normal distribution, you compare the characteristics of the data with the theoretical properties of the normal distribution or construct a normal probability plot
• For some variables, the descriptive characteristics of the data are inconsistent with the properties of the normal distribution
• Stem-and-leaf display or boxplot for small datasets, histogram for large datasets.
• Are the mean and median equal?
• Is interquartile range approximately 1.33 times the standard deviation? Is range 6 times stdev?
• Evaluate of the values are distributed. Do 2/3 of the values (68%) lie between mu +/- 1 sigma? Do 4/5 of the values lie between mu +/- 1.28? Do 19/20 of the values (95.5%) lie between mu +/- 2 sigma? 99.7% of values lie within 3 sigma.
• Normal probability plot: Shows whether data is left/right skewed or normal
• Quantile-quantile plot: Value on the y axis, z value on the x axis

Chapter 7: Sampling Distributions

• Trying to make inferences that are based on statistics calculated from samples
• Sample mean (statistic) estimates a population mean (parameter)
• Sample proportion (statistic) estimates the population proportion (parameter)
• Reach conclusion about the POPULATION not the sample
• Sampling distribution: The Distribution of results if you actually selected all possible samples. The single result you obtain in practice is just one of the results in the sampling distribution.

7.2: Sampling distribution of the mean

• The distribution of all possible sample means if you select all possible samples of a given size
• The sample mean is unbiased estimator of the population mean because the mean of all the possible sample means (of given size n) is equal to the population mean
• Standard error of the mean expresses how sample means vary from sample to sample.
• As the sample size increases, the standard error of the mean decreases by a factor equal to the square root fo the sample size
• Central Limit Theorem: Sample size of 30 prodces normal distribution of means no matter what the population distribution is.