《远大前程》英文介绍
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For any random variable X and value a, we can define the event A that X = a
Pr(A) = Pr(X=a) = Pr({x S| X(x)=a})
Two Coins Tossed
X: {TT, TH, HT, HH} → {0, 1, 2} counts # of heads
For any event A, can define the indicator random variable for A: XA(x) = 1 if x A 0 if x A
1
0.55
0.05 0 0.3 0.2
0.05 0.1 0.3
0
0.45
Definition: Expectation
It’s a function on the sample space S
It’s a variable with a probability distribution on its values
You should be comfortable with both views
From Random Variables to Events
Two Coins Tossed
X: {TT, TH, HT, HH} → {0, 1, 2} counts the number of heads Distribution on the HH S reals 2
¼ ¼ TT TH HT ¼ ¼ 0 1
¼
¼
½
It’s a Floor Wax And a Dessert Topping
Notational Conventions
Use letters like A, B, E for events
Use letters like X, Y, f, g for R.V.’s
R.V. = random variable
Two Views of Random Variables
Think of a R.V. as
X(3,4) = 3, Y(3,4) = 7, W(3,4) = 34, X(1,6) = 1 Y(1,6) = 7 Y(1,6) = 16 Y = sum of values of the two dice
W = (value of white die)value of black die
Tossing a Fair Coin n Times
COMPSCI 102
Introduction to Discrete Mathematics
CPS 102
Classics
Today, we will learn about a formidable tool in probability that will allow us to solve problems that seem really really messy…
On average, in class of size m, how many pairs of people will have the same birthday?
k k Pr(exactly k collisions) = k k (…aargh!!!!…)
The new tool is called “Linearity of Expectation”
S = all sequences of {H, T}n
D = uniform distribution on S D(x) = (½ )n for all x S Random Variables (say n = 10) X = # of heads X(HHHTTHTHTT) = 5 Y = (1 if #heads = #tails, 0 otherwise) Y(HHHTTHTHTT) = 1, Y(THHHHTTTTT) = 0
Random Variable
Let S be a sample space in a probability distribution
A Random Variable is a real-valued function on S Examples: X = value of white die in a two-dice roll
Pr(X = a) = Pr({x S| X(x) = a})
S
HH ¼ ¼ ¼ ¼
X
¼
2 0
TT
TH HT
¼
½
1
Pr(X = 1) = Pr({x S| X(x) = 1})
Distribution on X
= Pr({TH, HT}) = ½
From Events to Random Variables
Random Variable
To use this new tool, we will also need to understand the concept of a Random Variable Today’s lecture: not too much material, but need to understand it well
If I randomly put 100 letters into 100 addressed envelopes, on average how many letters wwenku.baidu.comll end up in their correct envelopes?
Hmm… k k Pr(k letters end up in correct envelopes) = k k (…aargh!!…)
Input to the function is random
A function from S to the reals
Or think of the induced distribution on Randomness is “pushed” to the values of the function
The expectation, or expected value of a random variable X is written as E[X], and is E[X] = Pr(x) X(x) = k Pr[X = k] xS k X is a function on the sample space S X has a distribution on its values
Pr(A) = Pr(X=a) = Pr({x S| X(x)=a})
Two Coins Tossed
X: {TT, TH, HT, HH} → {0, 1, 2} counts # of heads
For any event A, can define the indicator random variable for A: XA(x) = 1 if x A 0 if x A
1
0.55
0.05 0 0.3 0.2
0.05 0.1 0.3
0
0.45
Definition: Expectation
It’s a function on the sample space S
It’s a variable with a probability distribution on its values
You should be comfortable with both views
From Random Variables to Events
Two Coins Tossed
X: {TT, TH, HT, HH} → {0, 1, 2} counts the number of heads Distribution on the HH S reals 2
¼ ¼ TT TH HT ¼ ¼ 0 1
¼
¼
½
It’s a Floor Wax And a Dessert Topping
Notational Conventions
Use letters like A, B, E for events
Use letters like X, Y, f, g for R.V.’s
R.V. = random variable
Two Views of Random Variables
Think of a R.V. as
X(3,4) = 3, Y(3,4) = 7, W(3,4) = 34, X(1,6) = 1 Y(1,6) = 7 Y(1,6) = 16 Y = sum of values of the two dice
W = (value of white die)value of black die
Tossing a Fair Coin n Times
COMPSCI 102
Introduction to Discrete Mathematics
CPS 102
Classics
Today, we will learn about a formidable tool in probability that will allow us to solve problems that seem really really messy…
On average, in class of size m, how many pairs of people will have the same birthday?
k k Pr(exactly k collisions) = k k (…aargh!!!!…)
The new tool is called “Linearity of Expectation”
S = all sequences of {H, T}n
D = uniform distribution on S D(x) = (½ )n for all x S Random Variables (say n = 10) X = # of heads X(HHHTTHTHTT) = 5 Y = (1 if #heads = #tails, 0 otherwise) Y(HHHTTHTHTT) = 1, Y(THHHHTTTTT) = 0
Random Variable
Let S be a sample space in a probability distribution
A Random Variable is a real-valued function on S Examples: X = value of white die in a two-dice roll
Pr(X = a) = Pr({x S| X(x) = a})
S
HH ¼ ¼ ¼ ¼
X
¼
2 0
TT
TH HT
¼
½
1
Pr(X = 1) = Pr({x S| X(x) = 1})
Distribution on X
= Pr({TH, HT}) = ½
From Events to Random Variables
Random Variable
To use this new tool, we will also need to understand the concept of a Random Variable Today’s lecture: not too much material, but need to understand it well
If I randomly put 100 letters into 100 addressed envelopes, on average how many letters wwenku.baidu.comll end up in their correct envelopes?
Hmm… k k Pr(k letters end up in correct envelopes) = k k (…aargh!!…)
Input to the function is random
A function from S to the reals
Or think of the induced distribution on Randomness is “pushed” to the values of the function
The expectation, or expected value of a random variable X is written as E[X], and is E[X] = Pr(x) X(x) = k Pr[X = k] xS k X is a function on the sample space S X has a distribution on its values