Transcript Counting Data

Counting Data
Here are a few experiments:
• Toss a coin ten times, count # of “heads”
• Toss two dice, count sum of “pips”
• Count # of sales receipts at Meijer’s at the end of
a day
• Measure the average height reached by a toy
rocket in fifteen launches.
• Measure the average speed of one pitcher in
nine pitches.
THE QUESTION:
What do these separate
experiments have in
common?
mmmm …….
Any takers?
THE ANSWER
The end result is always
A NUMBER !
RANDOM VARIABLES
Quite clearly one cannot predict what number
one gets from an experiment. Than number
therefore can be said to vary at random. We
join the two concepts (variation and
randomness) in a double term:
Random Variable (RV for short)
RVs can be thought of as functions that attach
a number to each simple outcome of the
experiment.
Types of RV’s
• If all the possible values of a RV can be
counted, we call that RV
discrete.
• Otherwise we say the RV is
continuous.
Let’s look at a few examples:
Discrete or Continuous ?
• Sum of pips when tossing two dice
DISCRETE
• # of “Heads” in three coin tosses
DISCRETE
• # of coin tosses before the first “H”
DISCRETE
• The time it will take one of you to finish
Thursday’s exam
CONTINUOUS
 Toss a coin ten times, count # of “heads”
Discrete
 Count # of sales receipts at Meijer’s at the end of a
day
Discrete
 Measure the average height reached by a toy
rocket in fifteen launches.
Continuous
 Measure the average speed of one pitcher in nine
pitches.
Continuous
 Number of tosses of one die before a 6 shows up.
Discrete
The Probability Distribution
of a Random Variable
Recall that
RVs can be thought of as functions that
attach a number to each simple outcome of
the experiment.
The collection of numbers that a RV can
achieve is called the range of the RV.
Let’s look at an example.
Toss two tetrahedra
• Here is the sample space of tossing
two tetrahedra and recording the sum
of the two landed-on faces.
• Let X be the RV “sum of the two faces.”
Then the range of X clearly is
{2, 3, 4, 5, 6, 7, 8}
And
P(2) = 1/16; P(3) = 2/16; P(4) = 3/16;
P(5) = 4/16 ; P(6) = 3/16; P(7) = 2/16;
P(8) = 1/16.
We summarize the above with a table:
Another Example
• Toss an unfair coin 4 times. The coin favors
“H” 6 to 4, that is P(H)=0.6.
• Let X = # of “Tails”. The range is clearly
{0, 1, 2, 3, 4}
We will learn later that
4
k
4 k
P(X  k)    (0 .4) (0 .6)
 k
k  0,1,2,3, 4
So we get the probability distribution table:
Distribution Table:
For discrete RVs a table or a formula
generally suffice to describe the
distribution, for continuous RV the
situation will require graphs and a little
geometry. We’ll do that later.
The Expected Value
• Here is the Probability Distribution Table
of a RV originating from some unspecified
experiment
Suppose you repeat the experiment 20
times.
•
•
•
•
Since “probability” really means
“expected relative frequency” you would
expect
3 to happen
4 times (0.20x20)
5 to happen
7 times (0.35x20)
6 to happen
6 times (0.30x20)
8 to happen
3 times (0.15x20)
If you add up all these 20 numbers and
then divide by 20 you get the mean
(average) of the results. But this
operation gives:
(follow closely)
3  4 5 7  6  6 8 3

20
4
7
6
3
3
5 
6 
8 

20
20
20
20
 3  0 .2 0  5  0 .3 5  6  0 .3 0  8  0 .1 5 
 3  p(3)  5  p(5)  6  p(6)  8  p(8)
By the way, the answer is 5.35
We call this number the “expected value”
of the random variable X and denote it by
 or E(X) .
Formal Definition
• Following the pattern suggested by the
previous example we give the definition:
Let
be the probability distribution table of a
random variable X. The expected value of
X, denoted by  = E(X), is the number
n
 xk pk
k 1
Final Remarks
Remark # 1.
Remember that
p 1 + p 2 + ...+ p n = 1
Remark # 2.
If in the equation
x 1 p 1 + x 2 p 2 + ...+ x np n  
all numbers but one are known, the
unknown can be computed.
The Advantages of RV’s
• For a mathematician or a statistician the
most important benefit from using RVs is
that:
1. Even though they are just probability
spaces, the simple outcomes are just
NUMBERS,
not coins or dice or babies or patients.
This leads us to the second advantage
2. Totally different real life situations may
give rise to RVs with identical or at least
similar probability distributions.
Here is an example of four separate situations that
end up with identical RVs.
1. An unfair coin favors Heads 7 to 3. Toss it five
times, count Heads.
2. A bag has 70 quarters and 30 dimes. Pick one
coin from the bag, record its monetary value, put
it back in the bag. Do this five times and add up
the amounts you got.
3. 30% of the daily visitors to the Sporcizia Clinic
develop a bacterial infection. Five patients came
in today. Count how many did not develop the
infection.
4. 30% of the students at Podunk U. cheat a little.
Of five students chosen at random, count the
honest ones.
Note that the events
Heads, Q uarters, No Infection, Honest
are events with the same probability 0.7, and in all
four of the examples we are counting how many
times the event happens in five attempts. This
leads to the probability distribution table:
where
5 
p k =   (0 .7 )k (0 .3 )5 - k k= 0 ,1 ,2 ,3 ,4 ,5
k
We will learn later that the pk ‘s add up to 1.
Another example
1. Nature favors girls 52 to 48. A couple decides to
keep having babies until a boy is born or they
have six girls, whichever happens first (in my
hometown there was a family of 12 girls and one
boy, spoiled rotten he ended up in jail). X is the
number of children in that family.
2. A lot of 1,000 teething rings made in Lower
Slobbovia is known to have 520 with excessive
lead concentration. You keep buying a teething
ring until you get a good one or you have bought
six. X counts how many rings you buy.
In both examples you end up with a RV X
whose probability distribution table is:
k- 1
Where
p k = (0 .5 2 )
6
(0 .4 8 )
k= 1 ,2 ,3 ,4 ,5
5
p 6 = (0 .5 2 ) + (0 .5 2 ) (0 .4 8 )
We will learn later that the pk ‘s add up to 1.
Special RVs
The lesson to be garnered from the previous
examples is that a particular probability
distribution table can cover a multitude of
seemingly different real life situations. So it
behooves us to study individual RVs, each
with its own model of a Probability
Distribution Table.
We say we are studying Random Variables,
but we are really studying Distributions!
What identifies a distribution ?
The distribution tables we have seen so far
all look like this:
To fill the table we need to know:
1.
The top row
2.
The bottom row
This boils down to
1.
The range of values
2.
The individual probabilities
Special RV’s (cont’d)
Let’s start easy. Here is a probability distribution
table, let’s fill it:
The top is
any number
The bottom must be
100%
(This RV is neither R nor V, it’s called a
constant )
A little harder:
The top is
any two numbers (usually 1 and 0)
The bottom is
any two p and q, both non negative,
with p + q = 1
This RV has a name, Bernoulli RV, we look at
when it happens.
Bernoulli Trials
The Bernoulli RV happens whenever the
experiment has just two possible numerical
outcomes. In fact, if the actual outcomes are
not numerical we can make them so by
calling one of them 1 and the other 0 (we
could also use 17 or 347, but it gets
cumbersome!)
In everyday practice the two outcomes are
called “success” (corresponding to 1) and
“failure” (corresponding to 0). Careful …
The two words “success” and “failure” do not
have to maintain their usual meaning, in
some experiment success may mean the
fish died, or the rocket exploded, or the
baby was a female (I am being sexist, just
for fun!)
In our context, success simply means the
outcome that corresponds to the value 1,
and failure means the outcome
corresponding to the value 0.