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COMP 170 L2
L18: Random Variables: Independence and VariancePage 1
COMP 170 L2
Outline
Independence of RVs
Distribution Functions of RVs
Independence between RVs
Expectation of product of independent RVs
Variance of RV
Definition and Examples
Additivity
Standard deviation
Central limit theorem
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Distribution Functions of RVs
P(X=k) viewed as a function of k: The distribution function of X.
In general
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Probability Weight and Distribution FunctionPage 4
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Distribution Functions of RVsPage 5
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Distribution Functions of RVsPage 6
For coin example
D(1, 9) = P(1 <= X <=9 ) ~= 1
COMP 170 L2
Outline
Independence of RVs
Distribution Functions of RVs
Independence between RVs
Expectation of product of independent RVs
Variance of RV
Definition and Examples
Additivity
Standard deviation
Central limit theorem
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Independence Between Events
E is independent of F
If P(E|F) = P(E)
The information that “Event F occurred” does not change the
probability of E
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Random Variable and Event
Given a rand variable, we can define many difference events
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Independence between RVsPage 10
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Independence between RVsPage 11
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Independence between RVsPage 12
No need to further check
P(X=0, Y=1) ?= P(X=0)P(Y=1)
P(X=1, Y=0) ? = P(X=1)P(Y=0)
P(X=1, Y=1) ?= P(X=1)P(Y=1)
COMP 170 L2
Independence between RVs
Draw two cards from a deck of 52 cards
X: number on first card
Y: number on second card
X and Y are independent when drawing with replacement
X and Y are not independent when drawing without replacement
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COMP 170 L2
Outline
Independence of RVs
Distribution Functions of RVs
Independence between RVs
Expectation of product of independent RVs
Variance of RV
Definition and Examples
Additivity
Standard deviation
Central limit theorem
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Independence and ExpectationPage 15
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Independence and ExpectationPage 16
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Illustrating Proof via ExamplePage 19
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Illustrating Proof via ExamplePage 20
COMP 170 L2
Outline
Independence of RVs
Distribution Functions of RVs
Independence between RVs
Expectation of product of independent RVs
Variance of RV
Definition and Examples
Additivity
Standard deviation
Sum of independent RVs
The Trend
Central limit theorem
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Variance of RVs
Probability starts with a process whose outcome is uncertain
Sample space: the set of all possible outcomes
A random variable (RV) is a function defined on the sample space
Different runs of the process might yield different outcomes
The RV might take different values in different runs
In other words, the value of RV varies across different runs
Some RVs vary more and some vary less
Number of heads in 1 coin flip
Number of heads in 10 coin flips
Number of heads in 100 coin flips
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Variance of RVs
Variance of an RV X
Measures how much it varies (across different runs of process)
Relative to the mean value
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Which RV vary the most?
Flip fair coin
Number of heads in 1 flip
Number of heads in 10 flips
Number of heads in 100 flips
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Variance
1/4
10/4
100/4
COMP 170 L2
Outline
Independence of RVs
Distribution Functions of RVs
Independence between RVs
Expectation of product of independent RVs
Variance of RV
Definition and Examples
Additivity
Standard deviation
Central limit theorem
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Calculating VariancePage 28
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Example 1: We have already seen
Number of heads in n flips
n=1
n=10
(x1+X2+…+X10)
n=100
(X1+X2+…+X100)
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Variance
1/4
10/4 = 10 * 1/4
100/4 = 100 * 1/4
Additivity is true here.
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Additivity is true here.
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Example 3 (Counter Example) Page 31
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Example 3 (Counter Example) Page 32
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Two LemmasPage 34
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An Application of Theorem 5.29
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A CorollaryPage 38
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Outline
Independence of RVs
Distribution Functions of RVs
Independence between RVs
Expectation of product of independent RVs
Variance of RV
Definition and Examples
Additivity
Standard deviation
Central limit theorem
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Standard Deviation
Standard deviation is another measure of how much a rv varies or
how much a distribution spread out
It is the square root of variance.
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Next page
Shows several distributions, variances and standard deviations
Highlights the differences between variance and standard
deviation
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Distributions, Variances, and Standard Deviations Page 41
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The examples on the previous page show
Standard deviation is a natural measure of “spread” of distribution
Variance is easier to manipulate mathematically. Will see this in
further study of probability theory and statistics.
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Distributions, Variances, and Standard Deviations
COMP 170 L2
Outline
Independence of RVs
Distribution Functions of RVs
Independence between RVs
Expectation of product of independent RVs
Variance of RV
Definition and Examples
Additivity
Standard deviation
Central limit theorem
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A Pattern
If we flip of a coin a large number of times,
“The number of heads” has bell-shaped distribution.
This phenomenon is not unique to coin flipping
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Test with .8 probability of getting correct answer
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Normal DistributionPage 48
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Recap: 13-05-2010
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Recap: 13-05-2010
E is independent of F
If P(E|F) = P(E)
The information that “Event F occurred” does not change the
probability of E
COMP 170 L2
Recap: 13-05-2010
Flip fair coin
Number of heads in 1 flip: Variance = 1/4
Number of heads in 10 flips: Variance = 10/4
Number of heads in 100 flips: Variance = 100/4
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Number of times until first success
Throw a fair die
How many times, on average, do you need to throw the die until
you see a 1?
P(getting 1 at each throw) = 1/6
Answer: 1/(1/6) = 6
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Number of times until first success
Throw a fair die
How many times, on average, do you need to throw the die until
you see a 2 and 3 in that order?
Expected number of throws to see 2: 6
After that, expected number of throws to see 3: 6
Answer: 6+6 = 12
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Number of times until first success
Throw a fair die
How many times, on average, do you need to throw the die until
you see a 2 and 3 , where the order does not matter?
P( get 2 or 3 in one throw ) = 1/3
Expected number throws until you see one of 2 or 3: 3
After that, P( get the other number in each throw) = 1/6
Expected number of throws until you see the other number: 6
Answer: 3+6 = 9
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Old Exam Question 1Page 57
P( at least one copy of T1 in 20 weeks)
= 1 – P( no copy of T1 in 20 weeks)
= 1 – P ( no T1 in week1 AND no T1 in week 2 AND …)
= 1- P(no T1 in week1) P(no T1 in week 2) …
=
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Can we do this?
P(all 10 types of toys in 20 weeks)
= P(copy of T1 in 20 weeks AND copy of T2 in 20 weeks AND …)
= P(copy of T1 in 20 weeks) P(copy of T2 in 20 weeks)…
NO, events not independent
Correct way
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P(all 10 types of toys in 20 weeks) = 1 – P(A)
Use inclusion-exclusion to calculate P(A)
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Old Exam Question 2
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