comp 170 l2 l17: random variables and expectation page 1
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COMP 170 L2
L17: Random Variables and ExpectationPage 1
COMP 170 L2
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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COMP 170 L2
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”
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COMP 170 L2
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
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Example 2Page 5
COMP 170 L2
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
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 VariablesPage 15
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Expectation ExamplePage 16
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Expectation and Average
Expectation ~= Average over many runs of process
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Expect the average
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COMP 170 L2
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 spacePage 20
Get the same result from sample space
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Expectation from Sample spacePage 21
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COMP 170 L2
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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COMP 170 L2
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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COMP 170 L2
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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Repeating Bernoulli Trials Until SuccessPage 39
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Number of Trials until First SuccessPage 40
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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 SuccessPage 42
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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
Uncertain because outcome of process is
Bernoulli RV:
Outcome of Bernoulli Trail (success of failure)
Binomial RV
# of successes in n independent Bernoulli trials
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