Probability Handout - The Center for Brains, Minds & Machines

Crash Course on Basic Statistics

Marina Wahl, marina.w4hl@

University of New York at Stony Brook

November 6, 2013

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Contents

1 Basic Probability

1.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.2 Probability of Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.3 Bayes¡¯ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2 Basic Definitions

2.1 Types of Data . . . . . . .

2.2 Errors . . . . . . . . . . .

2.3 Reliability . . . . . . . . .

2.4 Validity . . . . . . . . . .

2.5 Probability Distributions .

2.6 Population and Samples .

2.7 Bias . . . . . . . . . . . .

2.8 Questions on Samples . .

2.9 Central Tendency . . . . .

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3 The Normal Distribution

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4 The Binomial Distribution

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5 Confidence Intervals

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6 Hypothesis Testing

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7 The t-Test

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8 Regression

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9 Logistic Regression

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10 Other Topics

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11 Some Related Questions

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CONTENTS

Chapter 1

Basic Probability

1.1

Basic Definitions

? The probability of an event is always between

0 and 1.

Trials

? The probability of an event and its complement

is always 1.

? Probability is concerned with the outcome of trials.

Several Events

? Trials are also called experiments or observations (multiple trials).

? The union of several simple events creates a

compound event that occurs if one or more

of the events occur.

? Trials refers to an event whose outcome is unknown.

? The intersection of two or more simple events

creates a compound event that occurs only if

all the simple events occurs.

Sample Space (S)

? Set of all possible elementary outcomes of

a trial.

? If events cannot occur together, they are mutually exclusive.

? If the trial consists of flipping a coin twice, the

sample space is S = (h, h), (h, t), (t, h), (t, t).

? If two trials are independent, the outcome of

one trial does not influence the outcome of another.

? The probability of the sample space is always

1.

Permutations

Events (E)

? Permutations are all the possible ways elements in a set can be arranged, where the order

is important.

? An event is the specification of the outcome of

a trial.

? An event can consist of a single outcome or a

set of outcomes.

? The number of permutations of subsets of size k

drawn from a set of size n is given by:

? The complement of an event is everything in

the sample space that is not that event (not E

or ¡« E).

nP k =

5

n!

(n ? k)!

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