Transcript Probability Review

Probability and Statistics Review
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Purpose:
– Review basics of probability and statistics
– Define some terminology
– Revisit some important distributions
– Discuss how to analyze and characterize
different probability distributions
– Discuss applicability to performance
evaluation (and CPSC 601.08)
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Some Terminology (1 of 2)
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Experiment (e.g., coin flipping)
Sample space (e.g., S ={Heads, Tails})
– Could be discrete or continuous
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Outcome (e.g., Heads)
Event: successful outcome occurs
Randomness: unpredictable outcomes
Independence: unaffected outcomes
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Some Terminology (2 of 2)
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Random variable X
Probability distributions
– Could be discrete or continuous
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Probability density function (pdf)
– f(x) = P(X = x)
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Cumulative Distribution Function (CDF)
– F(x) = P(X < x)
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CDF is integral of pdf (continuous case)
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Axioms of Probability
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Probabilities are non-negative
– For any event A in the sample space S,
P(A) > 0
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Probabilities are normalized
– P(S) = 1
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Mutually exclusive events
– If A and B are mutually exclusive events,
then P(A or B) = P(A) + P(B)
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Describing Distributions (1 of 2)
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There are several ways to summarize the
key properties of a distribution:
– Central tendency: mean, median, mode
– Variability: variance, standard deviation,
coefficient of variation (CoV), squared CoV
– Moments: 1st moment, 2nd moment, …
– Central moments: 1st central moment, …
– Modality, index of dispersion, skewness,
kurtosis, variance coefficient, …
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Describing Distributions (2 of 2)
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The most common summary statistics are
the mean and the variance:
– Mean: expected value (expectation)
– Variance: mean squared deviation from mean
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Mean is equal to the first moment
Variance can be calculated from the first
moment and the second moment
Variance is equal to 2nd central moment
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Some Common Distributions
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Uniform Distribution
Binomial Distribution
Geometric Distribution
Poisson Distribution
Exponential Distribution
Erlang Distribution
Gaussian (Normal) Distribution
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