Probability, Statistics, and Stochastics: Understanding Probabilities and Distributions, Assignments of Mathematics

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November 8, ’07 - Slide # 1

Probability, Statistics, and beyond ...

Igor Cialenco, IIT

[email protected], Office E1 234C

IIT, November 8, 2007

Summary First

November 8, ’07 - Slide # 2

Stochastics is fun to do !!!

Stochastics is important for many

real-life applications

Flip a coin

November 8, ’07 - Slide # 4

I

Flip a

fair

coin

I

All possible results or

outcomes

are

H

ead,

T

ail.

Sample Space

H, T

I

Every outcome has equal chances to happen (the coin is fair!).

I

So, the chances are

for

H, T

I

50% chances for

H

, and 50% for

T

I

In formulas

P

H

P

T

1 2

Fair Die

November 8, ’07 - Slide # 5

B

A fair die

P

k

1 6

, for

k

B

What are the chances (probabilities) of getting an even number?

B

P

3 6

1 2

B

P

(not to have 2

P

(to get 1

4 6

2 3

B

In general:if the outcomes are equally likely then for any

event

A

P

A

elements in

A

of total possible outcomes

Independence

November 8, ’07 - Slide # 7

B

If the event

B

does not affect the probability of event

A

, then

independent

B

P

A

P

A

B

or

P

A

and

B

P

A

P

B

B

Ex: 3 coins.

A

the first coin is

H

B

the third coin is

T

Independent.

B

HW:

Find some sets

A, B

from the above example such that

A, B

are dependent

B

Ex: A box with color balls, say 5 Green and 3 Blue.

Unfair Coin. General Probability

November 8, ’07 - Slide # 8

B

What can we do if the coin is not a fair one? How to detect thatthe coin is not fair? How unfair is it?

B

Specify the probabilities!

B

P

H

p

, then

P

T

p

B

In general

P

is a probability if:

i) for every

A

, P

A

ii)

p

iii) for every events

A

and

B

such that

A

B

(disjoint

events),

P

A

B

P

A

P

B

Back to flipping a coin

November 8, ’07 - Slide # 10

B

H, T

. The random variable

X

H

, X

T

B

P

X

p, P

X

p

. Bernoulli.

B

The Histogram, The Distribution ...

B

Theoretical vs Real-Life Repeated Outcomes

B

X

1

, X

2

,... , X

n

• independent Bernoulli(p)

B

Y

X

1

X

2

X

n

• Binomial(n,p)

Flip

n

coins and count the number of heads

B

P

Y

k

n k

p

k

p

n

−

k

, where

n k

n

!

k

!(

n

−

k

)!

http://members.shaw.ca/ron.blond/TLE/Bin.APPLET/index.htmlhttp://zoonek2.free.fr/UNIX/48_R/07.html

• Math - November 8, ’07 - Slide #
  • Lecture
    • November 15,

Other Discrete Distributions

November 8, ’07 - Slide # 13

Geometric distribution

p

P

X

k

p

p

k

−

1

k

Waiting time until first success in Bernoulli(

p

) independent trials

Negative Binomial

r, p

P

X

r

k

k

r

p

r

p

k

−

r

k

r, r

Waiting time for

r

-th success.

HW: Suppose that the number of typographical errors on a single page of a givenbook has a Poisson distribution with parameters

λ

=

1 2

. Find the probability that

there are at least two errors on page #10 (the book has more than 10 pages).HW: An urn contains

N

white and

M

black balls. Balls are randomly selected, one

at a time, with replacement, until a black one is obtained. What is the probabilitythat exactly

n

draws are needed.

hint: use Geometric(

p

). the problem is to guess

p

.

Continuous Distributions

November 8, ’07 - Slide # 14

Suppose

X

takes all real values

R

. It does not make any mathematical

sense to define

P

X

x

for all

x

R

Specify a function

f

X

x

, called the density function such that

f

R

f

x

dx

and put

P

X

a, b

b

a

f

X

x

dx

How does

f

look like? Almost the

histogram (scaled). Uniform distribution

a, b

f

x

b

a

for

x

[

a, b

]

and

otherwise X

∼ N

μ, σ

2

, called

Normal Distribution

with mean

μ

and standard

deviation

σ

if

f

X

x

πσ

e

−

(

x

−

μ

)

2

2

σ

2

N

• standard normal.

Examples

November 8, ’07 - Slide # 16

Bernoulli

p

E

X

p,

Var(

X

p

p

Binomial

n, p

E

X

np,

Var(

X

np

p

Geometric

p

E

X

p

−

1

Var(

X

p

−

2

p

Poisson

λ

E

X

λ,

Var(

X

λ

Negative Binomial

r, n

E

X

np

−

1

Var(

X

np

−

2

p

N

μ, σ

E

X

μ,

Var(

X

σ

2

Why Normal?

November 8, ’07 - Slide # 17

Central Limit Theorem Let

X

1

, X

2

be a sequence of i.i.d. random variables, each with

mean

μ

and variance

σ

2

. Then

X

1

X

2

X

n

nμ

σ

n

→ N

n

The convergence is in distribution, i.e.

P

∑

ni

=

X

i

n

μ

σ/

n

z

π

z

−∞

e

−

x

2

2

dx

See the histogram.Run the applet.

http://www.stat.sc.edu/~west/javahtml/CLT.html

Wiki CLT, links to some simulations

http://en.wikipedia.org/wiki/Central_limit_theorem

See the PDF’s from Matlab simulations

How to find

and

November 8, ’07 - Slide # 19

Suppose that

x

1

, x

2

,... , x

n

are

n

realizations of

X

(a population). Then

μ

n

k

=

x

n

n

μ

μ

• population mean;

μ

the real mean or sample mean. WHY?

Law of Large Numbers If

X

1

,... , X

n

i.i.d. Then,

1 n

n

i

=

X

i

μ,

n

Similarly, the approximation for the standard deviation

σ

2

n

n

i

=

x

2 i

μ

σ

2

Stochastic Processes. Random Walk

November 8, ’07 - Slide # 20

X

i

takes values

and

with probability

p

and

p

. Independent.

S

n

S

0

n

i

=

X

i

S

n

−

1

X

n

S

n

• Random Walk. if

p

• symmetrical random walk.

Absorbing barrier or Gambler’s ruin A Jaguar costs $

N

, and a gambler has an initial wealth of $

k

, with

0

< k < N

. The gambler plays with a banker the following game: he tosses a

coin (could be unfair) repeatedly; if the coin comes up head the banker pays$1, if tail the gambler looses $1. The game ends if the gambler has enoughmoney to buy a Jaguar or he is bankrupted. Find the probability that he isultimately bankrupted.