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COMP 170 L2
Page 1
L17: Random Variables and Expectation
COMP 170 L2
Page 2
Outline


Random Variables

The concept

Bernoulli/Binomial random variables
Expectation of Random Variables

The concept

Properties

Expectation and counting

Geometric distribution
COMP 170 L2
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Functions on Sample Space

Probability starts with a process (test, experiment) whose outcome is
uncertain

Sample space: the set of all possible outcomes

Sometimes, we want to define functions on the sample space

Example

Process: flip a coin n times

Sample space: set of sequences of n elements, each being H or T

Example: n=5, Function: “Number of heads”
COMP 170 L2
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Random Variables

Probability starts with a process whose outcome is uncertain

Sample space: the set of all possible outcomes

Sometimes, we want to define functions on the sample space

Outcome of the process is uncertain

Function defined on the outcome is also uncertain

So, called random variable

A random variable is a function defined on the sample space

Example: “Number of heads” in n coin flips is a random variables
COMP 170 L2
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Example 2
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Outline


Random Variables

The concept

Bernoulli/Binomial random variables
Expectation of Random Variables

The concept

Properties

Expectation and counting

Geometric distribution
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Binomial Random Variable

n Bernoulli trials, each with probability of success at each trial being p

X: number of successes,

X: called Binomial random variable

P: Binomial probability distribution

Do the numbers sum to 1?
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COMP 170 L2
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Outline


Random Variables

The concept

Bernoulli/Binomial random variables
Expectation of Random Variables

The concept

Properties

Expectation and counting

Geometric distribution
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Expectation of Random Variables
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Expectation Example
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Expectation and Average

Expectation ~= Average over many runs of process
Expect the average
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COMP 170 L2
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Outline


Random Variables

The concept

Bernoulli/Binomial random variables
Expectation of Random Variables

The concept

Properties
 Proof of linearity
 Use of linearity

Expectation and counting

Geometric distribution
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Expectation from Sample space

Get the same result from sample space
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Expectation from Sample space
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COMP 170 L2
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Outline


Random Variables

The concept

Bernoulli/Binomial random variables
Expectation of Random Variables

The concept

Properties
 Proof of linearity
 Use of linearity

Expectation and counting

Geometric distribution
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Outline


Random Variables

The concept

Bernoulli/Binomial random variables
Expectation of Random Variables

The concept

Properties

Expectation and counting

Geometric distribution
COMP 170 L2
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COMP 170 L2
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Outline


Random Variables

The concept

Bernoulli/Binomial random variables
Expectation of Random Variables

The concept

Properties

Expectation and counting

Geometric distribution
COMP 170 L2
Repeating Bernoulli Trials Until Success
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Number of Trials until First Success
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
Let x = 1-p

Consider only 0 < p < 1

Then, 0 < x <1

Let n goes to infinity, LHS becomes
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Number of Trial until First Success
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COMP 170 L2
Recap: 06-05-2010

Probability starts with a process whose outcome is uncertain

Sample space: the set of all possible outcomes

A random variable is a function defined on the sample space


Bernoulli RV:


Uncertain because outcome of process is
Outcome of Bernoulli Trail (success of failure)
Binomial RV

# of successes in n independent Bernoulli trials
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