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Basic Statistical ConceptsCRSC- SAMSI Workshops
Jiezhun (Sherry) GuDepartment of Statistics
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What is statistics?
Statistics is the science of making inference based on the data.
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What is sample space?
The set S, of all possible outcomes of a particular experiment is called the sample space for the experiment.
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What is an event?
An event is any collection of possible outcomes of an experiment, that is, any subset of S (including S itself).
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Frequentist interpretation of probability: Probability is the language used to capture certain phenomena with unpredictable outcomes, such phenomena are said to be random. Bayesian interpretation of probability: degree of plausibility that an event will occur.
What’s probability?
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What is random variable?
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Property of random variable
continuity:
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Property of random variableIndependency:
(1) Two events A and B are independent iff P(A ∩B)=P(A)P(B)(2) Two random variables X and Y are independent iff for any numbers a and b the events [X ≤ a] and [Y ≤ b] are independent events as defined above. (3)Similarly an arbitrary collection of random variables -- possible more than just two of them -- is independent precisely if for any finite collection X1, ..., Xn and any finite set of numbers a1, ..., an, the events [X1 ≤ a1], ..., [Xn ≤ an] are independent events as defined above.
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Property of random variable
Univariate or Multivariate
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Probability function on random variable:
Probability density function, when random variable is continuous.Probability mass function, when random variable is discrete.
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DEF: Populations and Samples
A data sample is taken from a population.
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DEF: Parameters & Statistics
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Statistical summary
Sample mean
Sample variance
2 2
1
1 ( )1
n
ii
S X Xn =
= −− ∑
1
1 n
ii
X Xn =
= ∑
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Statistical summary
A confidence interval is an interval constructed about the parameter estimate in a manner such that (1-α) % of all samples will produce an interval containing the true parameter.
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What is Model?
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Deterministic Models
Deterministic models are fixed; same inputs always result in the same output.
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Stochastic Models
Stochastic models have random output based on the probability distribution of the random variable.
Parametric statistical model
Nonparametric statistical model
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Normal distribution
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Standard Normal
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Central Limit theorem
As the sample size gets very large, the distribution of the sample means becomes normal.
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What statistics Can Do?
If we know our model, we can make predictions about future outcomes.
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What statistics Can Do?
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What statistics Can Do?
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What statistics Can Do?
Confidence Interval- Large Sample
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What statistics Can Do?
Confidence Interval- Small Sample
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Thank You