Review of Exam I Sections 2.2 -- 4.5 - PowerPoint Presentation

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Review of Exam I

Sections 2.2 -- 4.5

Jiaping WangDepartment of Mathematical Science 02/18/2013, Monday

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Outline

Sample Space and Events

Definition of Probability Counting Rules

Conditional Probability and Independence

Probability Distribution and Expected Values

Bernoulli, Binomial and Geometric Distributions

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Part

1. Sample Space and Events

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Definition 2.1

A

sample space S

is a set that includes all possible outcomes for a random experiment

listed in a

mutually exclusive

and

exhaustive way.Mutually Exclusive means the outcomes of the set do not overlap.Exhaustive means the list contains all possible outcomes.

Definition 2.2:

An

event

is any subset of a sample space.

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There are three operators between events:

Intersection: ∩ --- A∩B or AB – a new event consisting of common elements from A and B

Union: U --- AUB – a new event consisting of all outcomes from A or B.Complement: ¯, A, -- a subset of all outcomes in S that are not in A.

Event Operators and Venn Diagram

AUB

A∩B

A

S

S

S

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Commutative laws:

Associate laws:

Distributive laws:

DeMorgan’s

laws:

Some Laws

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Part

2. Definition of Probability

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Suppose that a random experiment has associated with a sample space S. A

probability

is a numerically valued function that assigned a number P(A) to every event A so that the following axioms hold:(1) P(A) ≥ 0

(2) P(S) = 1

(3) If A

1

, A

2, … is a sequence of mutually exclusive events (that is Ai∩Aj=ø for any i≠j

), then

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2. 0≤ P(A) ≤1for any event A.

3. P(AUB) = P(A) + P(B) if A and B are mutually exclusively.Some Basic Properties

1. P( ø ) = 0, P(S) = 1.5. If A is a subset of B, then P(A) ≤ P(B).

4. P(

AUB) = P(A) + P(B) – P(A∩B) for general events A and B.

6. P(A) = 1 – P(A).

7. P(A∩B) = P(A) – P(A∩B).

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Theorem 2.1. For events A

1, A2, …, An

from the sample space S,

We can use induction to prove this.

Inclusive-Exclusive Principle

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Determine the Probability Values

The definition of probability only tells us the axioms that the probability function must obey; it doesn’t tell us what values to assign to specific event.

For example, if a die is balanced, then we may think P(Ai)=1/6 for Ai={ i },

i = 1, 2, 3, 4, 5, 6

The value of the probability is usually based on empirical evidence or on careful thought about the experiment.

However, if a die is not balanced, to determine the probability, we need run lots of experiments to find the frequencies for each outcome.

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Part

3. Counting Rules

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Fundamental Principle of Counting:

If the first task of an experiment can result in n1 possible outcomes and for each such outcome, the second task can result in n

2 possible outcomes, then there are n1n2 possible outcomes for the two tasks together.

Theorem 2.2

The principle can extend to more tasks in a sequence.

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Order Is

Important

Order Is Not ImportantWith Replacementnr

Crn+r-1

Without Replacement

P

r

n CrnOrder and Replacement

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Theorem 2.5 Partitions

Consider a case: If we roll a die for 12 times, how many possible ways to have

2 1’s, 2 2’s, 3 3’s, 2 4’s, 2 5’s and 1 6’s? Solution: First, choose 2 1’s from 12 which gives 12!/(2!10!), second, since there aretwo positions are filled by 1’s, the next choice appears in the left 10 positions, so there are 10!/(8!2!) ways, and so similar for next other selections which provides the final result is 12!/(2!10!)x10!/(2!8!)x8!/(3!5!)x5!/(2!3!)x3!/(2!1!)x1!/(1!0!) =12!/(2!x2!x3!x2!x2!x1!)

Theorem 2.5 Partitions.

The number of partitioning n distinct objects into k groups containing n1

, n

2

,•••, nk objects, respectively, is

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Part

4. Conditional Probability and Independence

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Definition 3.1

If A and B are any two events, then the conditional probability of A given B, denoted as P(A|B), is

Provided that P(B)>0.

Notice that P(A∩B) = P(A|B)P(B) or P(A∩B) = P(B|A)P(A).

This definition also follows the three axioms of probability.

A

∩B is a subset of B, so P(A∩B )≤P(B), then 0≤P(A|B)≤1;P(S|B)=P(S∩B)/P(B)=P(B)/P(B)=1;

If A1, A

2

, …, are mutually exclusively, then so are A

1

∩B, A

2

∩B, …; and

P(UA

i

|B) = P((UA

i

) ∩B)/P(B)=P(U(A

i

∩B)/P(B)=∑P(A

i

∩B)/P(B)= ∑P(A

i

|B).

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Theorem 3.2: Multiplicative Rule. If A and B are any two events, then

P(A∩B) = P(A)P(B|A) = P(B)P(A|B)If A and B are independent, then P(A∩B) = P(A)P(B).

Definition 3.2 and Theorem 3.2

Definition 3.2: Two events A and B are said to be independent if

P(A∩B)=P(A)P(B).This is equivalent to stating that

P(A|B)=P(A), P(B|A)=P(B)

If the conditional probability exist.

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Theorem of Total Probability:

If B

1

, B

2

, …, B

k

is a collection of mutually exclusive and exhaustive events, then for any event A, we have

Bayes

’ Rule. If the events B1, B2, …, Bk form a partition of the sample space S, and A is any event in S, then

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Part

5. Probability Distribution and Expected Value

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A random variable X is said to be

discrete

if it can take on only a finite number – or a countably infinite number – of possible values x. The probability function of X, denoted by p(x), assigns probability to each value x of X so that the following conditions hold: P(X=x)=p(x)≥0;

∑ P(X=x) =1, where the sum is over all possible values of x.

A random variable is a real-valued function whose domain is a sample space.

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The

distribution function

F(b) for a random variable X is F(b)=P(X ≤ b);If X is discrete,

Where p(x) is the probability function. The distribution function is often called

the cumulative distribution function (CDF).

Any function satisfies the following 4 properties is a distribution function:

1. 2.

3.

The distribution function is a non-decreasing function: if a<b, then F(a)≤ F(b). The distribution function can remain constant, but it can’t decrease as we increase from a to b.

4. The distribution function is right-hand continuous:

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Definition 4.4 The expected value of a discrete random variable X with probability distribution p(x) is given as

(The sum is over all values of x for which p(x)>0)

We sometimes use the notation E(X)=μfor this equivalence.

Definition 4.4

Note: Not all expected values exist, the sum above must converge absolutely,

∑|

x|p

(x)<∞.

Theorem 4.1 If X is a discrete random variable with probability p(x) and if g(x) is any real-valued function of X, then

E(g(x))=∑g(x)p(x).

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Definitions 4.5 and 4.6

The variance of a random variable X with expected value

μ is given by V(X)=E[(X- μ)

2]

Sometimes we use the notation

σ

2 = E[(X- μ)2]For this equivalence.

The standard deviation is a measure of variation that maintains the original units of measure.

The standard deviation of a random variable is the square root of the variance and is given by

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Theorem 4.2 For any random variable X and constants a and b.

E(

aX + b) = aE(X) + b V(aX

+ b) = a2

V(X)

Standardized random variable: If X has mean

μ

and standard deviation σ, then Y=(X – μ)/

σ

has E(Y)=0 and V(Y)=1, thus Y can be called the standardized random variable of X.

Theorem 4.3 If X is a random variable with mean

μ

, then

V(X)= E(X

2

) –

μ

2

Tchebysheff’s

Theorem. Let X be a random variable with mean

μ

and standard deviation

σ

.

Then for any positive k,

P(|X –

μ

|/

σ

< k) ≥ 1-1/k

2

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Part

6. Bernoulli, Binomial and Geometric Distribution

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Let the probability of success is p, then the probability of failure is 1-p, the distribution of X is given by

p(x)=px(1-p)1-x, x=0 or 1Where p(x) denotes the probability that X=x.E(X) = ∑

xp(x) = 0p(0)+1p(1)=0(1-p)+p= p

 E(X)=p

V(X)=E(X

2

)-E2(X)= ∑x2p(x

) –p2=0(1-p)+1(p)-p

2

=p-p

2

=p(1-p)



V(X)=p(1-p)

Bernoulli Distribution

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Binomial Distribution

Suppose we conduct n independent Bernoulli trials, each with a probability p of success. Let the random variable X be the number of successes in these n trials. The distribution of X is called binomial distribution.

Let Yi = 1 if ith

trial is a success = 0 if

ith trial is a failure, Then X

=∑

Y

i denotes the number of the successes in the n independent trials.So X can be {0, 1, 2, 3, …, n}.For example, when n=3, the probability of success is p, then what is the probability of X?

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Cont.

From the binomial formula,

we can have

=

 

The mass function of binomial distribution:

 

A random variable X is a binomial distribution if

1. The experiment consists of a fixed number n of identical trials.

2. Each trial only have two possible outcomes, that is the

B

ernoulli trials.

3. The probability p is constant from trial to trial.

4. The trials are independent.

5. X is the number of successes in n trails

.

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E(X)=

np

Bernoulli random variables Y1, Y2, …, Yn, then

 

V(X)=

np

(1-p)

Bernoulli random variables Y

1

, Y

2

, …, Y

n

, then

V

 

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The geometric distribution function:

P(X=x)=p(x)=(1-p)xp

=qxp, x= 0, 1, 2, …., q=1-p

Geometric Distribution: Probability Function

P(X=x) =

q

x

p = p[qx-1

p] = qP(X=x-1)

<P(X=x-1)

as q ≤ 1, for x=1, 2, …

A Geometric Distribution Function with p=0.5

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Geometric Series and CDF

The geometric series: {t

x: x=0, 1, 2, …}

Sum of Geometric series: For |t|<1, we have

=

Sum of partial series

:

=

 

Then we can verify

 

The cumulative distribution function:

F(x)=P(

X≤x

)=

=

=1-q

x+1

And P(

X≥x

)=1-F(x-1)=q

x

 

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Mean and Variance

E(X)=

So E(X)/(

pq

) =

And E(X)/p = [0 + q + 2q

2

+

…

]

Thus, E(X)/(

pq

)-E(X)/p = 1+q+q2+q3+ • • • = 1/(1-q)

 E(X)=

 

The Expected Value

E(X)=

The Variance

V(X)=