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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