Probability & Independence. Sample space Random variable Probability.

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Presentation transcript:

Probability & Independence

Sample space Random variable Probability

face Y f 1 1/ Random variable

Sample point ( ): the object where selection is made Sample space ( ) : the set of all sample points

Random variable (,, …. ): a characteristic (number, color, etc.) of sample points. a function converts sample points to their characteristics. a function from sample space to the set of real numbers.

Random variables No.ColorEven…f 00Green-1/38 0Green-1/38 1RedX1/38 2BlackO1/38 3RedX1/38 4BlackO1/38 35BlackX1/38 36RedO1/38

Random variable generates subsets of the sample space. Event: a subset of the sample space

A B C E Each point in layout corresponds to an event.

Probability is a function converts events to a number between 0 and 1.

: complementary event of : sample space, the total set, the universal event : null event

: the set of even numbers : multiples of 3 : multiples of 6 : multiples of 2 or 3

: sample space, the total set, universal event, roulette, chocolate box, urn : set of subsets of layout

RouletteHolesCharacteristics of holes sample points, sample spacerandom variables Layout set of subsets of sample space

X f 1 1/ (X,Y) f (1,1)1/36 (1,2)1/36 (1,3)1/36 … (6,5)1/36 (6,6)1/36

How to get money from casino ?

Bet $100 + “all the amount you lose” every time. Think coin tossing game. Then you will win at least a time, then stop there.

$500 House marginUsing chipsBetting limit

“Who are willing to play a gambling game, seeing rising sun?”

How to get money from casino ? The best strategy is the one that casinos want to keep out of. That is the strategy leaving casino as soon as possible.

Counterplots of casino: High quality accommodation Far away location

Independence Combinatorics Joint distribution Conditional distribution

(X,Y) f (1,1)1/36 (1,2)1/36 (1,3)1/36 … (6,5)1/36 (6,6)1/36 X\Y 12…6Tot 1 1/36 1/6 2 1/36 1/6 … … … 6 1/36 1/6 Tot1/6 1.0 Joint dist’n, Marginal dist’n

X\Y123456Tot 1(H) 2/18 2/3 0(T) 1/18 1/3 Tot 1/6 1.0 X\Y123456Tot 1(H) 0(T) Tot 1.0

Independence of random variables X\Y123456Tot 1(H) 2/18 2/3 0(T) 1/18 1/3 Tot 1/6 1.0 X\Y123456Tot 1(H) 2/181/182/181/182/181/181/2 0(T) 1/182/181/182/181/182/181/2 Tot 1/6 1.0

joint pdf, marginal pdf, independence X\Y 012T 0 1/161/81/161/4 1 1/81/41/81/2 2 1/161/81/161/4 T 1/21/41.0 X\Y 012T 0 1/8 01/4 1 1/81/41/81/2 2 01/8 1/4 T 1/21/41.0 X\Y 012T 0 01/ /2 2 01/40 T 1/21/41.0

conditional distribution f(x|y) X\Y 012T 0 1/161/81/161/4 1 1/81/41/81/2 2 1/161/81/161/4 T 1/21/41.0 X\Y 012T 0 1/8 01/4 1 1/81/41/81/2 2 01/8 1/4 T 1/21/41.0 X\Y /4 1 1/2 2 1/4 T 1.0 X\Y /21/40 1 1/2 2 01/41/2 T 1.0

X\Y 012T 0 1/161/81/161/4 1 1/81/41/81/2 2 1/161/81/161/4 T 1/21/41.0 X\Y 012T 0 1/41/21/ /41/21/ /41/21/41.0 X\Y /4 1 1/2 2 1/4 T 1.0

X\Y 012T 0 1/161/81/161/4 1 1/81/41/81/2 2 1/161/81/161/4 T 1/21/41.0 X\Y 012T 0 1/8 01/4 1 1/81/41/81/2 2 01/8 1/4 T 1/21/41.0 X\Y 012T 0 01/ /2 2 01/40 T 1/21/41.0

XYX Yf1f1 f 2 f3f3 0001/161/ /4 0201/ /8 1/ /8 1/4 2001/ /8 1/4 2241/161/80 Total1.0 E(XY)15/41 Cov(X,Y)01/40

X

(X)(X) (0)(0)

(o)(o) (X)(X) regardless of independence

?

a a b How many ways to give an order to n people ?

How many ways to give an order to r people selected from n people ? --- No order for (n-r) people. Give an full order to n people, and disregard the order of the last (n-r) people.

How many ways to separate n people into two groups of r people and (n-r) people ? Disregard also the order of selected r people.

X: the number of heads when we toss a fair coin twice Xf Total1.0

Thank you !!