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- +(or -) covariance, if not iid
- The signage related to covariance is the same signage for the left side of the equation.
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What do we mean by the term statistical inference?
Taking sample data and relating it back to a population with mathematical rigor
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Define a discrete random variable
has a countable support (can list the values taken on)
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Define a continuous random variable
expected outcomes (support) is infinite (has an interval or union of intervals as the support)
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What is the most important thing that knowing a distribution allows us to find?
Probabilities!
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What is meant by the term 'random sample'?
Independently and identically distributed (iid)
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What does assuming a random sample do for us (in terms of a joint distribution?
- joint distributions are the product of marginal distributions
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What is a statistic?
A statistic is a function of RVs (no unknowns) from a random sample
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What are large sample theory approximations?
- independence
- identically distributed (not always assumed)
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What is a sampling distribution?
- distribution of a statistic
- pattern and frequency the sample mean is observed
- the larger the sample size, the sooner the distribution of the samples converges on the true distribution of the population
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What is the usefulness of understanding a large-sample distribution of a statistic?
- approximating probability about our statistic
- also, large sample approximations make our life easier
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Central Limit Theorem
- The central limit theorem says that the sampling distribution of the mean will always be normally distributed, as long as the sample size is large enough.
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What is the purpose of Delta Method Normality?
- Using Delta Method Normality, we can often extend the large-sample Normality to other functions of averages
- works because of Taylor series Expansion
- Linear expansion of a normal distribution is still normal
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Formal notation for Large Sample Normality and Delta Method:
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What is the (weak) Law of Large Numbers?
- Big picture goal is to estimate parameters such as μ
- Convergence in probability to a population mean
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Continuity theorem, as related to sample variance:
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Making Inference on μ using Large-Sample Results:
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Z-score related to 95% confidence
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What theorem allows us to substitute a 'consistent' estimator of σ ?
Slutsky's theorem!
by CLT: (in distribution)
by continuity: , (in probability)
by Slutsky: (in probability)
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Non-technically, what do we mean by the term distribution?
The distribution describes the different possible outcomes of a random variable, and the frequency at which they occur
- To be inclusive of continuous functions:
- a function that defines the relationship between outcomes and probabilities
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Which functions uniquely characterize a distribution?
- PDF/PMF: probability (density/mass) function
- CDF: cumulative density function
- MGF: Moment generating function
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What is: mean, variance, skewness, kertosis?
- mean: center of distribution, the first moment
- variance: the expected value of the squared deviation from the mean of a random variable
- skewness: measurement of the distortion of symmetrical distribution or asymmetry in a data set
- kertosis: a measure of the "tailedness" of the probability distribution of a real-valued random variable
note: you can have a distribution with all the same primary moments, but doesn't have the same distribution. So these can describe a distribution, but not characterize it
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Name the distribution:
"# of correct answers in 20 questions on an exam"
Binomial
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Name the distribution:
"# of times your parents call before you pick up"
Geometric
(stops when you pick up the phone)
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Name the distribution:
"# of good parts selected from a lot of parts where 3/4 are good"
hypergeometric
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Name the distribution:
"# of times you look at your phone in an hour"
Poission
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Name the distribution:
"Average cost of books for a sample of 100 NC State students"
Normal
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Name the distribution:
"Time until failure for a part"
gamma
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What is the 'sampling distribution' of the 'sample mean'?
The distribution that describes the different changes in sample means for each sample
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