Probability and Statistics
Kristel Van Steen, PhD2
Montefiore Institute - Systems and Modeling
GIGA - Bioinformatics
ULg
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CHAPTER 5: SAMPLING TO APPROXIMATE THE TRUE WORLD
1 Introduction
2 Generating data
2.1 Design of experiments
2.2 Sampling designs and towards inference
2.3 Ethics
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3 Sample versus population
3.1 Introduction
3.2 Distribution of a sample
3.3 Statistics and sample moments
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4 Sample mean
4.1 Mean and variance
4.2 Law of large numbers revisited
4.3 Central-limit theory revisited
4.4 Bernoulli and Poisson distribution
4.5 Exponential distribution
4.6 Uniform distribution
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5 Sampling from the normal distribution
5.1 The role of normal distributions in statistics
5.2 Sample mean
5.3 The chi-square distribution
5.4 The F distribution
5.5 The Student’s t distribution
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6 Future highlights
6.1 Estimating parameters
6.2 Order statistics
6.3 Sample size calculations
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1 Introduction
Probability theory versus statistics
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Inductive versus deductive inference
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Inductive versus deductive inference
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2 Generating data
2.1 Design of experiments
Obtaining data
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Population versus sample
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Observational studies versus experiments
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Some terminology
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Comparative experiments
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Placebo effects
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Caution about experimentation
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Other ways to remove bias
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Lack of realism
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Randomization
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Principles of experimental design
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Completely randomized designs
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Block designs
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Matched pairs designs
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Why experimental designs ?
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2.2 Sampling designs and towards inference
Sampling methods
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Simple random samples
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Stratified samples
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Caution about sampling surveys
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2.3 Ethics
Institutional Review Boards
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Informed Consent
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Confidentiality
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Clinical trials
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Behavioral and social science experiments
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3 Sample versus population
3.1 Introduction
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Towards statistical inference
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Sampling variability
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3.2 Distribution of a sample
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3.3 Statistics and sample moments
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Practical note
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Capture-recapture sampling
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4 Sample mean
4.1 Mean and variance
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For normally distributed populations
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On a practical note
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4.2 Law of large numbers revisited
• Question: using only a finite number of values of X (a random sample
of size n, say), can any reliable inference be made about E[X], “the
average of an infinite number of values of X”?
• Answer: YES!
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• A positive integer n can be determined such that if a random sample
of size n or larger is taken from a population with density f(.) (with
E[X])=µ), the probability can be made as close to 1 as desired that the
sample mean X bar will deviate from µ by less than an arbitrarily
specified small quantity:
(cfr. weak law of large numbers)
• Proof: Use Chebyshev inequality
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4.3 Central-limit theory revisited
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How large a sample size ?
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4.4 Bernoulli and Poisson distribution
See before
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Binomial distributions for sample counts
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Binomial distribution in statistical sampling
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Reminder: sampling variability
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Binomial mean and standard deviation
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Sample proportions
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Normal approximation
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Sampling distribution of the sampling proportion
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Normal approximation: continuity correction
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4.5 Exponential distribution
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4.5 Exponential distribution
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4.6 Uniform distribution
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on
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5 Sampling from the normal distribution
5.1 The role of normal distributions in statistics
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5 Sampling from the normal distribution
5.1 The role of normal distributions in statistics
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5.2 Sample mean
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5.3 The chi-square distribution
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square distribution
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• Recall: that if two moment generating functions both exi
equal (they agree), then the corresponding cumulative distribution
functions are the same (agree)
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Recall: that if two moment generating functions both exi
equal (they agree), then the corresponding cumulative distribution
functions are the same (agree)
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Recall: that if two moment generating functions both exist and are
equal (they agree), then the corresponding cumulative distribution
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5.4 The F distribution
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(**)
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(**)
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This theorem can be very useful in sampling
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This theorem can be very useful in sampling
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5.5 The Student’s t distribution
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dent’s t distribution
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(^^)
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(^^)
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(^^)
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