Lecture 5

LECTURES 5-6: Simple Statistics I-II

Statistics relevant for AMS321: Probability

* Permutation:

To select r elements out of a total of n distinct elements. Any particular arrangement, or order, is called Permutation.

Now figure out the #Permutations.

1st selection, on all n objects

2nd selection, on n-1 objects

r-th selection, on n-(r-1) objects

By Thm 3.1 (multiplication rule),

we have n*(n-1)*...*[n-(r-1)] Permutations.

P(n,r)	= n(n-1)...*[n-(r-1)]
	= n(n-1)...[n-(r-1)][n-r]...321/[n-r]...321
	= n!/(n-r)!

P(n,r) = n!/(n-r)!

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| Thm 3.2: P(n,r) = n!/(n-r)! |
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Remember 0! == 1 by definition.

e.g.1, order 2 out of 3 students. how many ways do you have?

P(3,2) = 3!/(3-2)! = 6

AB, AC, BC, BA, CA, CB,

e.g.2

How many ways do you have to order 3 out of 3 objects.

P(3,3) = 3!/(3-3)! = 6

ABC, ACB,

BAC, BCA,

CAB, CBA

e.g.3

How many ways do you have to order n out of n objects.

P(n,n) = n!/(n-n)! = n!

* Combinations ---- we no longer care about the orders.

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|C(n,r) = (n,\r) = n!/r!(n-r)!|
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e.g.1, select 2 out of 3 computer. how many ways do you have?

C(3,2) = 3!/2!(3-2)! = 3.

AB, Ac, BC.

BA, cA, CB (these are equivalent to above.)

e.g.2

How many ways do you select 3 out of 3 objects.

C(3,3) = 3!/3!(3-3)! = 1

ABC

e.g.3

How many ways do you have to select n out of n objects.

C(n,n) = n!/n!(n-n)! = 1

* Probability:

Def: Among "n" equally likely possibilities, the number of times that certain outcome regarded as "success" occur is "s". The probability of success is s/n.

e.g.1

What's the probability of drawing an ace from a well shuffled deck of 52 cards?

We know there are s=4 aces and n=52 , So, s/n=4/52 = 1/13.

* Defs:

Probability: possible outcome experiment

sample space: all possible outcomes of an experiment

Event: a subset of a sample space

discrete sample space: finite or countable infinite elements

continuous sample space: continuous elements

e.g. Certain airlines offer reward for people who can suspend over-crowded flights. Here is a sample space that the Smiths (husband and wife) will get the two rewards:

(1, 1) = (Mr. got it, Mrs. got it.)

S={(0,0), (1,0), (2,0), (0,1), (0,2) (1,1)}

---A total of six outcomes.

A = {(1,0), (0,1)}

B = {(0,0), (1,1)}

mutually exclusive events: nothing in common

union, intersection, complement,

Two subsets in a sample space: A U B is union (A or B or both)

draw pictures

A n B is the intersect (A and B)

A' = Complement A (everything in S that's not in A)

* Counting:

Tree diagram: counting is very tedious, a tree diagram helps.

e.g. The outcome for a course:

professsor: good, average, bad.

contents: interesting, boring

student: working hard, so-so, ignoring

professors

  ------  ----

         ----    --

--

--

---

---

there are 18 different outcomes.

Thm 3.1:

If sets A1, A2, ...Ak contains n1, n2, ...nk elements, there are n1*n2*...*nk ways to select first one element from A1, and then A2... and Ak.

e.g.1 in our example above, n1=3, n2=2, n3=3, so, there are n1*n2*n3=18 ways.

e.g.2 there are 10 true-false questions in a test, how many different answers can one get?

n1= 2, n2=2, ..., n10=2

So, there are n1*n2*...n3=2^10 = 1024 ways.

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Normal distribution(physicists call it Gaussian distribution)

1. draw a picture for the distribution

2. prove it's normalized.

3. standard normal distribution:

Now, in the normal distribution:

if we set

	= f(z)

Then,

After this, if we want to compute the

4. obviously,

e.g. The actual amount of coffee from a vending machine sold to a 4-ounch jars is considered as a normal random variable with standard deviation

If only 2% jars contains less than 4 ounce. What should be the mean?

Remember you can deliver more, but you can't deliver less.

Solution:

(Remember F(z) + F(-z) = 1)

After looking up the table, we get

thus,

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Normal approximation to binomial distribution:

Recall binomial distribution can be written as

Thm 5.1

If X follows b(X; n, p). If n-> and then the limiting form of the distribution function of this standardized random variable as n--> is given by

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Uniform distribution:

f(x) = 1/(b-a), x in (a,b)

Mean:

Variance:

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Log-normal distribution:

for x>0 and b>0

(See exact formula at the bottom of p155.)

This function describes a distribution that the log of a variable has normal distribution.

normal distribution

normal (gaussian, bell-shaped) distribution

standard normal (gaussian, bell-shaped) distribution