Transcript Randomness and Probability

Business Statistics for Managerial
Decision
Randomness and Probability
The Idea of Probability
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Toss a coin, or choose a SRS. The result can not
be predicted in advance, because the result will
vary when you toss the coin or choose the sample
repeatedly.
But there is still a regular pattern in the results, a
pattern that emerges only after many repetitions.
Chance behavior is unpredictable in the short run
but has a regular and predictable pattern in the
long run.
This fact is the basis for the idea of probability.
The Idea of Probability
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The proportion of tosses
of a coin that give a head
changes as we make more
tosses.
Eventually , however, the
proportion approaches 0.5,
the probability of a head.
This figure shows the
results of two trials of
5000 tosses.
Randomness and Probability
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We call a phenomenon random if individual
outcomes are uncertain but there is
nonetheless a regular distribution of
outcomes in a large number of repetition.
The probability of an outcome of a random
phenomenon is the proportion of times the
outcome would occur in a very long
repetitions.
Probability Models
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A probability model is a mathematical description
of a random phenomenon consisting of two parts:
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The sample space of a random phenomenon is the
set of all possible outcomes.
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A sample space S
A way of assigning probabilities to events.
S is used to denote sample space.
An event is an outcome or a set of outcomes of a
random phenomenon.
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An event is a subset of the sample space.
Example, Rolling Dice
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There are 36 possible
outcomes when we
roll two dice and
record the up-faces in
order(first die, second
die).
They make up the
sample space S.
Probability Rules
1.
2.
3.
4.
The probability p(A) of any event A satisfies
0  p( A)  1
If S is the sample space in a probability model, then
p(S) =1.
The probability that an event A does not occur is
p( A does not occur) = 1- P(A)
Two event A and B are disjoint if they have no outcomes
in common and so can never occur simultaneously.
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If A and B are disjoint,
P(A or B) = P(A) + P(B)
Venn Diagram
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Venn diagram showing
disjoint events A and
B
P( A or B)  P( A)  P( B)
P( A  B)  P( A)  P( B)
Venn Diagram
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Venn diagram showing
events A and B that
are not disjoint.
The event {A and B}
consists of outcomes
common to A and B.
P( A or B)  P( A)  P( B)  P( A and B)
P( A  B)  P( A)  P( B)  P( A  B)
Example
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Recall the 36 possible outcomes of rolling
two dice. What probabilities shall we assign
to these outcomes?
What is the probability of rolling a 5?
What is the probability of rolling a 7?
What is the probability of rolling a seven or
eleven?
Assigning Probabilities: Finite Number of
Outcomes
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Assign probabilities to each individual
outcome.
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These probabilities must be numbers between 0
and 1.
They must have sum 1.
The probability of any event is the sum of
the probabilities of the outcomes making up
the event.
Probability Histograms
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We can use histograms to display probability
distributions as well as distribution of data.
In a probability histogram the height of each bar
shows the probability of the outcome at its base
Since the heights are probabilities, they add to 1
As usual the bars in a histogram have the same
width, therefore, the areas also display the
assignment of probability outcomes.
Think of these histograms as idealized pictures of
the results of very many trials.
Example
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You are asked to select a non zero digit
randomly.
The sample space for this experiment is:
The probability model for this experiment is
completely defined by this table.
X
Probability
1
1/9
2
1/9
3
1/9
4
1/9
5
1/9
6
1/9
7
1/9
8
1/9
9
1/9
Example
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The probability that
the first digit is equal
to or grater than 6
is:_________
The probability that
the first digit is grater
than 6 is:___________
1.00
0.80
probability
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0.60
0.40
0.20
0.00
0
1
2
3
4
5
Outcome
6
7
8
9
Example
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Probability histogram
for random digits 1 to
9
The height of each bar
shows the probability
assigned to a single
outcome.
Assigning Probabilities: Intervals of
Outcomes
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Suppose you are asked to select a number 0 and at
random. What is the sample space?
The sample space is:
S = { all numbers between 0 and 1}
For example: 0.2, 0.27, .00387, etc
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Call the outcome of this example (the number you
select) Y for short.
How can we assign probabilities to such events as
p(.3  y  .7)?
Assigning Probabilities: Intervals of
Outcomes
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We need a new way of assigning
probabilities to events - as areas under a
density curve.
Recall we first introduced density curves as
models for data in previous lectures.
A density curve has area exactly 1
underneath it, corresponding to total
probability 1.
Example
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Probability as area
under a density curve
These uniform density
curves spread
probability evenly
between 0 and 1.
Example
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Probability as area
under a density curve
These uniform density
curves spread
probability evenly
between 0 and 1.
Normal Probability Models
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Any density curve can be used to assign
probabilities.
The density curves that are most familiar to
us are the normal curves introduced in the
previous lectures.
Normal distributions are probability models.
Example
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The weights of all 9-ounce bags of a particular
brand of potato chip, follow the normal
distribution with mean  = 9.12 ounces and
standard deviation  = 0.15 ounces, N(9.12, 0.15).
Let’s select one 9-ounce bag at random and call its
weight W.
What is the probability that it has weights between
9.33 and 9.45 ounces?
Example
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The probability in the
example as an area
under the standard
normal curve.
Random Variables
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Not all sample spaces are made up of numbers.
When we toss a coin four times, we can record the
outcome as a string of heads and tails, such as HTTH.
However we are most often interested in numerical
outcomes such as the count of heads in the four tosses.
It is convenient to use the following shorthand notation
 Let x be the number of heads.
 If our outcome is HTTH, then X = 2, if the next
outcome is TTTH, the value of X changes to 1.
Random Variables
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The possible values of X are 0, 1, 2, 3, 4.
Tossing a coin four times will give X one of
these possible values.
We call X a random Variable because its
values vary when the coin tossing is
repeated.
The random digit example used this
shorthand notation.
Random Variables
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In the potato chip example, we let W stand
for the weight of a randomly selected 9ounce bag of potato chips.
We know that W would take a different
value if we took another random sample.
Because its value changes from one sample
to another, W is also a random variable.
Random Variables
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A random Variable is a variable whose value is a
numerical outcome of a random phenomenon.
We usually denote random variables by capital
letters, such as X, Y.
The random variable of greatest interest to us are
outcomes such as the mean X of a random
sample, for which we keep the familiar notation.
Random Variables
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There are two types of random variables
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A discrete random variable has finitely many
possible values.
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Discrete
Continuous
Random digit example
A continuous random variable takes all values in
some interval of numbers.
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Random numbers between 0 and 1 example.
Probability Distribution
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The starting point for studying any random
variable is its probability distribution, which is just
the probability model for the outcomes.
The probability distribution of a random variable
X tells us what values X can take and how to
assign probabilities to those values.
Since the nature of sample spaces for discrete and
continuous random variables are different, we
describe probability distributions for the two types
of random variables separately.
Discrete Probability Distributions
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The probability distribution of a discrete random
variable X lists the possible values of X and their
probabilities:
Value of X
Probability
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x2
p2
x3
p3
…
…
xk
pk
The probabilities pi must satisfy two requirements.
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x1
p1
Every probabilities pi is a number between 0 and 1.
The sum of the probabilities is exactly 1
To find the probability of any event, add the probabilities
pi of the individual values xi that makes up the event.
Example
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Buyers of a laptop computer model may choose to
purchase either 10 GB, 20 GB, 30 GB or 40 GB
internal hard drive. Choose customers from the
last 60 days at random to ask what influenced their
choice of computer. To “choose at random” means
to give every customer of the last 60 days the
same chance to be chosen. The size of the internal
hard drive chosen by a randomly selected
customer is a random variable X.
Example
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The value of X changes when we repeatedly
choose customers at random, but it is always one
of 10, 20, 30, or 40 GB. The probability
distribution of X is
Hard drive X
10
20
30
40
probability
.50
.25
.15
.10
The probability that a randomly selected customer
chose at least a 30 GB hard drive is:
Example
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We can use a probability histogram to display a discrete
distribution.
The following probability histogram pictures this
distribution.
Example: four coin tosses
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Toss a balanced coin four times; the discrete
random variable X counts the number of heads.
How shall we find the probability distribution of
X?
The outcome of four tosses is a sequence of heads
and tails such as HTTH.
There are 16 possible outcomes.
The following figure lists the outcomes along with
the value of X for each outcome.
Example: four coin tosses
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Possible outcomes in four tosses of a coin.
The Random Variable X is the number of heads.
Example: four coin tosses
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The probability of each value of X can be
found using the previous figure as follows:
1
 .0625
16
4
 1) 
 .25
16
6
 2) 
 .375
16
4
 3) 
 .25
16
1
 4) 
 .0625
16
p ( X  0) 
p( X
p( X
P( X
P( X
Example: four coin tosses
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These probabilities have sum=1, so this is a
legitimate probability distribution.
In the table form, the distribution is
Number of heads X
Probability
0
.0625
1
.25
2
.375
3
.25
4
.0625
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The probability of tossing at least two heads is:
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The probability of at least one head is:
Example: four coin tosses
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Probability histogram
for the number of
heads in four tosses of
a coin
Continuous Probability Distribution
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A continuous random variable like uniform random
number Y between 0 and 1 or the normal package weight
W of potato chips has an infinite number of possible
values.
Continuous probability distribution therefore assign
probabilities directly to events as area under a density
curve.
The probability distribution of a continuous random
variable X is described by a density curve.
The probability of any event is the area under the density
curve and above the values of X that make up the event.
Continuous Probability Distribution
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The probability distribution for a continuous
random variable assigns probabilities to
intervals of outcomes rather than to
individual outcomes.
All continuous probabilities assign
probability 0 to every individual outcome.
Example
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The actual tread life of a 40,000-mile automobile
tire has a Normal probability distribution with  =
50,000 and  = 5500 miles. We say X has a
N(50000, 5500) distribution. The probability that a
randomly selected tire has a tread life less than
40,000 mile
p( X  40000)  p(
 P( Z  1.82)
 .0344
x  50000 40000  50000
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)
5500
5500
Example
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The normal distribution
with  = 50,000 and
 = 5500.
The shaded area is
P(X < 40000).
Can a Continuous Random Variable be
exactly 0.8?
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This figure shows the
uniform density curve for
outcomes between 0 and
1.
Probabilities are the areas
under this curve.
What is
P(.79 < Y < .81)?
What is
P(.799 < Y < .801)?
What is P(Y = .8)?
Can a Continuous Random variable be
exactly 0.8?
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Since P(Y = 0.8) = 0 then the two events
{Y > 0.8} and {Y  0.8} have the same
probability.
We ignore the distinction between > and 
when finding probabilities for continuous
(but not discrete) Random Variables.
The Mean of a Random Variable
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We can speak of the mean winning in a game of
chance or the standard deviation of randomly
varying number of calls a travel agency receives in
an hour.
The mean X of a set of observation is their
ordinary average.
The mean of a random variable X is also the
average of the possible values of X, but in this
case not all outcomes need to be equally likely.
Mean of a Discrete Random Variable
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Suppose that X is a discrete random
variable whose distribution is
Value of X
Probability
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x1
p1
x2
p2
…
…
x3
p3
xk
pk
To find the mean of X, multiply each
possible value by its probability, then add
all the products:
  x p  x p  x p   x p
x
1 1
k
  xi pi
i 1
2
2
3
3
k
k
Hard-Drive Example
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The following table gives the distribution of
customer choices of hard-drive size for a
laptop computer model.Find the mean of
this probability distribution.
Hard drive X
probability
10
.50
20
.25
30
.15
40
.10
 x  10  .50  20  .25  30  .15  40  .10  18.5
Rules for the Mean
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Rule 1: If X is a random variable and a and b are
fixed numbers, then
a bX  a  b  x
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Rule 2: If X and Y are random variables, then
 X Y   X  Y
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This is the addition rule for means
Example: Portfolio Analysis
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The past behavior of
two securities in
Sadie’s portfolio is
pictured in this figure,
which plots the annual
returns on treasury
bills and common
stocks for years 1950
to 2000.
Example: Portfolio Analysis
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We have calculated the mean returns for these data
set.
 X = annual return on T-bills  X  5.2%
 Y = annual return on stocks Y  13.3%
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Sadie invests 20% in T-bills, and 80% in common
stocks. Find the mean expected return on her
portfolio.
R  .2 X  .8Y
 R  .2   X  .8  Y
 .2  5.2  .8 13.3  11.68%
Variance of a Discrete Random
Variable
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Suppose that X is a discrete random variable
whose distribution is
Value of X
Probability
x1
p1
x2
p2
x3
p3
…
…
xk
pk
and that  is the mean of X. The variance of X
 X2  ( x1   X ) 2 p1  ( x2   X ) 2 p2  ( x3   X ) 2 p3    ( xk   X ) 2 pk
k
  ( xi   ) 2 pi
i 1
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The standard deviation X of X is the square root of the variance.
Rules for the Variance
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Rule 1: If X is a random Variable and a and b are
fixed numbers, then
 a2bX  b 2 X2
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Rule 2: If X and Y are independent random
Variables, then 2
2
2
 X Y   X   Y
 X2 Y   X2   Y2
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This is the addition rule for variances of the
independent random variables.
Rules for the Variance
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Rule3: If X and Y have correlation , then
 X2 Y   X2   Y2  2 X  Y
 X2 Y   X2   Y2  2 X  Y
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This is the general addition rule for
variance of random variables.
Example: Portfolio Analysis
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Based on annual returns between 1950 and 2000,
we have
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X = annual return on T-bills x = 5.2% X = 2.9
Y = annual return on stocks Y = 13.3% Y = 17%
Correlation between x and Y:
 = - 0.1
For the return R on the Sadie’s portfolio of 20% Tbill and 80% stocks,
R  .2 X  .8Y
 R  .2   X  .8  Y
 .2  5.2  .8 13.3  11.68%
Example: Portfolio Analysis
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To find the variance of the portfolio return, combine Rules
1 and 3.
 R2   .22 X   .28Y  2  .2 X  .8Y
 (.2) 2  X2  (.8) 2  Y2  2  (.2   X )(. 8   Y )
 (.2) 2 (2.9) 2  (.8) 2 (17) 2  2(0.1)(. 2  2.9)(. 8  17)
 183.719
 R  183.719  13.55%
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The portfolio has a smaller mean return than all-stock portfolio,
but it is also less volatile.
Mean of a Continuous Random
Variable
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The probability distribution of a continuous
random variable X is described by a density curve.
Recall that the mean of the distribution is the point
at which the area under the density curve would
balance if it were made out of solid material.
The mean lies at the center of symmetric density
curves such as the Normal curve.
Exact calculation of the mean of a distribution
with a skewed density curve requires advanced
mathematics.
Mean of a Continuous Random
Variable
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The idea that the mean is the balance point of the
distribution applies to discrete random variables as
well, but in the discrete case we have a formula
that gives us the numerical value of .
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Mean and variance rules holds for mean and
variance of both discrete and continuous random
variables