Title: 6.2 Probability Theory Longin Jan Latecki Temple University
16.2 Probability TheoryLongin Jan LateckiTemple
University
- Slides for a Course Based on the TextDiscrete
Mathematics Its Applications (6th Edition)
Kenneth H. Rosen based on slides by - Michael P. Frank and Andrew W. Moore
2Terminology
- A (stochastic) experiment is a procedure that
yields one of a given set of possible outcomes - The sample space S of the experiment is the set
of possible outcomes. - An event is a subset of sample space.
- A random variable is a function that assigns a
real value to each outcome of an experiment
Normally, a probability is related to an
experiment or a trial.
Lets take flipping a coin for example, what are
the possible outcomes?
Heads or tails (front or back side) of the coin
will be shown upwards.
After a sufficient number of tossing, we can
statistically conclude that the probability of
head is 0.5.
In rolling a dice, there are 6 outcomes. Suppose
we want to calculate the prob. of the event of
odd numbers of a dice. What is that probability?
3Random Variables
- A random variable V is any variable whose value
is unknown, or whose value depends on the precise
situation. - E.g., the number of students in class today
- Whether it will rain tonight (Boolean variable)
- The proposition Vvi may have an uncertain truth
value, and may be assigned a probability.
4Example 10
- A fair coin is flipped 3 times. Let S be the
sample space of 8 possible outcomes, and let X be
a random variable that assignees to an outcome
the number of heads in this outcome. - Random variable X is a function XS ? X(S),
where X(S)0, 1, 2, 3 is the range of X, which
is the number of heads, andS (TTT), (TTH),
(THH), (HTT), (HHT), (HHH), (THT), (HTH) - X(TTT) 0 X(TTH) X(HTT) X(THT) 1X(HHT)
X(THH) X(HTH) 2X(HHH) 3 - The probability distribution (pdf) of random
variable X is given by P(X3) 1/8, P(X2)
3/8, P(X1) 3/8, P(X0) 1/8.
5Experiments Sample Spaces
- A (stochastic) experiment is any process by which
a given random variable V gets assigned some
particular value, and where this value is not
necessarily known in advance. - We call it the actual value of the variable, as
determined by that particular experiment. - The sample space S of the experiment is justthe
domain of the random variable, S domV. - The outcome of the experiment is the specific
value vi of the random variable that is selected.
6Events
- An event E is any set of possible outcomes in S
- That is, E ? S domV.
- E.g., the event that less than 50 people show up
for our next class is represented as the set 1,
2, , 49 of values of the variable V ( of
people here next class). - We say that event E occurs when the actual value
of V is in E, which may be written V?E. - Note that V?E denotes the proposition (of
uncertain truth) asserting that the actual
outcome (value of V) will be one of the outcomes
in the set E.
7Probabilities
- We write P(A) as the fraction of possible worlds
in which A is true - We could at this point spend 2 hours on the
philosophy of this. - But we wont.
8Visualizing A
Event space of all possible worlds
P(A) Area of reddish oval
Worlds in which A is true
Its area is 1
Worlds in which A is False
9Probability
- The probability p PrE ? 0,1 of an event E
is a real number representing our degree of
certainty that E will occur. - If PrE 1, then E is absolutely certain to
occur, - thus V?E has the truth value True.
- If PrE 0, then E is absolutely certain not to
occur, - thus V?E has the truth value False.
- If PrE ½, then we are maximally uncertain
about whether E will occur that is, - V?E and V?E are considered equally likely.
- How do we interpret other values of p?
Note We could also define probabilities for more
general propositions, as well as events.
10Four Definitions of Probability
- Several alternative definitions of probability
are commonly encountered - Frequentist, Bayesian, Laplacian, Axiomatic
- They have different strengths weaknesses,
philosophically speaking. - But fortunately, they coincide with each other
and work well together, in the majority of cases
that are typically encountered.
11Probability Frequentist Definition
- The probability of an event E is the limit, as
n?8, of the fraction of times that we find V?E
over the course of n independent repetitions of
(different instances of) the same experiment. - Some problems with this definition
- It is only well-defined for experiments that can
be independently repeated, infinitely many times!
- or at least, if the experiment can be repeated in
principle, e.g., over some hypothetical ensemble
of (say) alternate universes. - It can never be measured exactly in finite time!
- Advantage Its an objective, mathematical
definition.
12Probability Bayesian Definition
- Suppose a rational, profit-maximizing entity R is
offered a choice between two rewards - Winning 1 if and only if the event E actually
occurs. - Receiving p dollars (where p?0,1)
unconditionally. - If R can honestly state that he is completely
indifferent between these two rewards, then we
say that Rs probability for E is p, that is,
PrRE p. - Problem Its a subjective definition depends on
the reasoner R, and his knowledge, beliefs,
rationality. - The version above additionally assumes that the
utility of money is linear. - This assumption can be avoided by using utils
(utility units) instead of dollars.
13Probability Laplacian Definition
- First, assume that all individual outcomes in the
sample space are equally likely to each other - Note that this term still needs an operational
definition! - Then, the probability of any event E is given by,
PrE E/S. Very simple! - Problems Still needs a definition for equally
likely, and depends on the existence of some
finite sample space S in which all outcomes in S
are, in fact, equally likely.
14Probability Axiomatic Definition
- Let p be any total function pS?0,1 such
that ?s p(s) 1. - Such a p is called a probability distribution.
- Then, the probability under p of any event E?S
is just - Advantage Totally mathematically well-defined!
- This definition can even be extended to apply to
infinite sample spaces, by changing ???, and
calling p a probability density function or a
probability measure. - Problem Leaves operational meaning unspecified.
15The Axioms of Probability
- 0 lt P(A) lt 1
- P(True) 1
- P(False) 0
- P(A or B) P(A) P(B) - P(A and B)
16Interpreting the axioms
- 0 lt P(A) lt 1
- P(True) 1
- P(False) 0
- P(A or B) P(A) P(B) - P(A and B)
The area of A cant get any smaller than 0
And a zero area would mean no world could ever
have A true
17Interpreting the axioms
- 0 lt P(A) lt 1
- P(True) 1
- P(False) 0
- P(A or B) P(A) P(B) - P(A and B)
The area of A cant get any bigger than 1
And an area of 1 would mean all worlds will have
A true
18Interpreting the axioms
- 0 lt P(A) lt 1
- P(True) 1
- P(False) 0
- P(A or B) P(A) P(B) - P(A and B)
19These Axioms are Not to be Trifled With
- There have been attempts to do different
methodologies for uncertainty - Fuzzy Logic
- Three-valued logic
- Dempster-Shafer
- Non-monotonic reasoning
- But the axioms of probability are the only system
with this property - If you gamble using them you cant be
unfairly exploited by an opponent using some
other system di Finetti 1931
20Theorems from the Axioms
- 0 lt P(A) lt 1, P(True) 1, P(False) 0
- P(A or B) P(A) P(B) - P(A and B)
- From these we can prove
- P(not A) P(A) 1-P(A)
- How?
21Another important theorem
- 0 lt P(A) lt 1, P(True) 1, P(False) 0
- P(A or B) P(A) P(B) - P(A and B)
- From these we can prove
- P(A) P(A B) P(A B)
- How?
22Probability of an event E
- The probability of an event E is the sum of the
probabilities of the outcomes in E. That is - Note that, if there are n outcomes in the event
E, that is, if E a1,a2,,an then -
23Example
- What is the probability that, if we flip a coin
three times, that we get an odd number of tails? - (TTT), (TTH), (THH), (HTT), (HHT), (HHH), (THT),
(HTH) - Each outcome has probability 1/8,
- p(odd number of tails) 1/81/81/81/8 ½
24Visualizing Sample Space
- 1. Listing
- S Head, Tail
- 2. Venn Diagram
- 3. Contingency Table
- 4. Decision Tree Diagram
25Venn Diagram
Experiment Toss 2 Coins. Note Faces.
Tail
Event
TH
HT
HH
Outcome
TT
S
Sample Space
S HH, HT, TH, TT
26Contingency Table
Experiment Toss 2 Coins. Note Faces.
nd
2
Coin
st
1
Coin
Head
Tail
Total
Outcome
SimpleEvent (Head on1st Coin)
Head
HH
HT
HH, HT
Tail
TH
TT
TH, TT
Total
HH,
TH
HT,
TT
S
S HH, HT, TH, TT
Sample Space
27Tree Diagram
Experiment Toss 2 Coins. Note Faces.
H
HH
H
T
HT
Outcome
H
TH
T
T
TT
S HH, HT, TH, TT
Sample Space
28Discrete Random Variable
- Possible values (outcomes) are discrete
- E.g., natural number (0, 1, 2, 3 etc.)
- Obtained by Counting
- Usually Finite Number of Values
- But could be infinite (must be countable)
29Discrete Probability Distribution ( also called
probability mass function (pmf) )
- 1. List of All possible x, p(x) pairs
- x Value of Random Variable (Outcome)
- p(x) Probability Associated with Value
- 2. Mutually Exclusive (No Overlap)
- 3. Collectively Exhaustive (Nothing Left Out)
- 4. 0 ? p(x) ? 1
- 5. ? p(x) 1
30Visualizing Discrete Probability Distributions
Table
Listing
Tails
f(x
)
p(x
)
- (0, .25), (1, .50), (2, .25)
Count
0
1
.25
1
2
.50
2
1
.25
p(x)
Graph
Equation
.50
n
!
x
n
x
?
p
x
p
p
(
)
(
)
?
?
1
.25
x
n
x
!
(
)
!
?
x
.00
0
1
2
31Arity of Random Variables
- Suppose A can take on more than 2 values
- A is a random variable with arity k if it can
take on exactly one value out of v1,v2, .. vk - Thus
32Mutually Exclusive Events
- Two events E1, E2 are called mutually exclusive
if they are disjoint E1?E2 ? - Note that two mutually exclusive events cannot
both occur in the same instance of a given
experiment. - For mutually exclusive events, PrE1 ? E2
PrE1 PrE2.
33Exhaustive Sets of Events
- A set E E1, E2, of events in the sample
space S is called exhaustive iff
. - An exhaustive set E of events that are all
mutually exclusive with each other has the
property that
34An easy fact about Multivalued Random Variables
- Using the axioms of probability
- 0 lt P(A) lt 1, P(True) 1, P(False) 0
- P(A or B) P(A) P(B) - P(A and B)
- And assuming that A obeys
35Another fact about Multivalued Random Variables
- Using the axioms of probability
- 0 lt P(A) lt 1, P(True) 1, P(False) 0
- P(A or B) P(A) P(B) - P(A and B)
- And assuming that A obeys
36Elementary Probability Rules
- P(A) P(A) 1
- P(B) P(B A) P(B A)
37Bernoulli Trials
- Each performance of an experiment with only two
possible outcomes is called a Bernoulli trial. - In general, a possible outcome of a Bernoulli
trial is called a success or a failure. - If p is the probability of a success and q is the
probability of a failure, then pq1.
38Example
- A coin is biased so that the probability of heads
is 2/3. What is the probability that exactly
four heads come up when the coin is flipped
seven times, assuming that the flips are
independent? - The number of ways that we can get four heads is
C(7,4) 7!/4!3! 75 35 - The probability of getting four heads and three
tails is (2/3)4(1/3)3 16/37 - p(4 heads and 3 tails) is C(7,4) (2/3)4(1/3)3
3516/37 560/2187
39Probability of k successes in n independent
Bernoulli trials.
- The probability of k successes in n independent
Bernoulli trials, with probability of success p
and probability of failure q 1-p is
C(n,k)pkqn-k
40Find each of the following probabilities when n
independent Bernoulli trials are carried out with
probability of success, p.
- Probability of no successes.
- C(n,0)p0qn-k 1(p0)(1-p)n (1-p)n
- Probability of at least one success.
- 1 - (1-p)n (why?)
41Find each of the following probabilities when n
independent Bernoulli trials are carried out with
probability of success, p.
- Probability of at most one success.
- Means there can be no successes or one success
- C(n,0)p0qn-0 C(n,1)p1qn-1
- (1-p)n np(1-p)n-1
- Probability of at least two successes.
- 1 - (1-p)n - np(1-p)n-1
42A coin is flipped until it comes ups tails. The
probability the coin comes up tails is p.
- What is the probability that the experiment ends
after n flips, that is, the outcome consists of
n-1 heads and a tail? - (1-p)n-1p
43Probability vs. Odds
ExerciseExpress theprobabilityp as a
functionof the odds in favor O.
- You may have heard the term odds.
- It is widely used in the gambling community.
- This is not the same thing as probability!
- But, it is very closely related.
- The odds in favor of an event E means the
relative probability of E compared with its
complement E. O(E) Pr(E)/Pr(E). - E.g., if p(E) 0.6 then p(E) 0.4 and O(E)
0.6/0.4 1.5. - Odds are conventionally written as a ratio of
integers. - E.g., 3/2 or 32 in above example. Three to two
in favor. - The odds against E just means 1/O(E). 2 to 3
against
44Example 1 Balls-and-Urn
- Suppose an urn contains 4 blue balls and 5 red
balls. - An example experiment Shake up the urn, reach in
(without looking) and pull out a ball. - A random variable V Identity of the chosen
ball. - The sample space S The set ofall possible
values of V - In this case, S b1,,b9
- An event E The ball chosen isblue E
______________ - What are the odds in favor of E?
- What is the probability of E?
b1
b2
b9
b7
b5
b3
b8
b4
b6
45Independent Events
- Two events E,F are called independent if
PrE?F PrEPrF. - Relates to the product rule for the number of
ways of doing two independent tasks. - Example Flip a coin, and roll a die.
- Pr(coin shows heads) ? (die shows 1)
- Prcoin is heads Prdie is 1 ½1/6 1/12.
46Example
Suppose a red die and a blue die are rolled. The
sample space
Are the events sum is 7 and the blue die is 3
independent?
47The events sum is 7 and the blue die is 3 are
independent
S 36
p(sum is 7 and blue die is 3) 1/36 p(sum is 7)
p(blue die is 3) 6/366/361/36 Thus, p((sum is
7) and (blue die is 3)) p(sum is 7) p(blue die
is 3)
48Conditional Probability
- Let E,F be any events such that PrFgt0.
- Then, the conditional probability of E given F,
written PrEF, is defined as PrEF
PrE?F/PrF. - This is what our probability that E would turn
out to occur should be, if we are given only the
information that F occurs. - If E and F are independent then PrEF PrE.
- ? PrEF PrE?F/PrF PrEPrF/PrF
PrE
49Visualizing Conditional Probability
- If we are given that event F occurs, then
- Our attention gets restricted to the subspace F.
- Our posterior probability for E (after seeing F)
correspondsto the fraction of F where Eoccurs
also. - Thus, p'(E)p(EnF)/p(F).
Entire sample space S
Event F
Event E
EventEnF
50Conditional Probability Example
- Suppose I choose a single letter out of the
26-letter English alphabet, totally at random. - Use the Laplacian assumption on the sample space
a,b,..,z. - What is the (prior) probabilitythat the letter
is a vowel? - PrVowel __ / __ .
- Now, suppose I tell you that the letter chosen
happened to be in the first 9 letters of the
alphabet. - Now, what is the conditional (orposterior)
probability that the letteris a vowel, given
this information? - PrVowel First9 ___ / ___ .
1st 9letters
vowels
w
z
r
k
b
c
a
t
y
u
d
f
e
x
g
i
o
l
s
h
j
n
p
m
q
v
Sample Space S
51Example
- What is the probability that, if we flip a coin
three times, that we get an odd number of tails
(event E), if we know that the event F, the
first flip comes up tails occurs? - (TTT), (TTH), (THH), (HTT), (HHT),
(HHH), (THT), (HTH) - Each outcome has probability 1/4,
- p(E F) 1/41/4 ½, where Eodd number of
tails - or p(EF) p(E?F)/p(F) 2/4 ½
- For comparison p(E) 4/8 ½
- E and F are independent, since p(E F) Pr(E).
52Prior and Posterior Probability
- Suppose that, before you are given any
information about the outcome of an experiment,
your personal probability for an event E to occur
is p(E) PrE. - The probability of E in your original probability
distribution p is called the prior probability of
E. - This is its probability prior to obtaining any
information about the outcome. - Now, suppose someone tells you that some event F
(which may overlap with E) actually occurred in
the experiment. - Then, you should update your personal probability
for event E to occur, to become p'(E) PrEF
p(EnF)/p(F). - The conditional probability of E, given F.
- The probability of E in your new probability
distribution p' is called the posterior
probability of E. - This is its probability after learning that event
F occurred. - After seeing F, the posterior distribution p' is
defined by letting p'(v) p(vnF)/p(F) for
each individual outcome v?S.
53 6.3 Bayes Theorem Longin Jan LateckiTemple
University
- Slides for a Course Based on the TextDiscrete
Mathematics Its Applications (6th Edition)
Kenneth H. Rosen based on slides by - Michael P. Frank and Wolfram Burgard
54Bayes Rule
- One way to compute the probability that a
hypothesis H is correct, given some data D - This follows directly from the definition of
conditional probability! (Exercise Prove it.) - This rule is the foundation of Bayesian methods
for probabilistic reasoning, which are very
powerful, and widely used in artificial
intelligence applications - For data mining, automated diagnosis, pattern
recognition, statistical modeling, even
evaluating scientific hypotheses!
Rev. Thomas Bayes1702-1761
55Bayes Theorem
- Allows one to compute the probability that a
hypothesis H is correct, given data D
Set of Hj is exhaustive
56Example 1 Two boxes with balls
- Two boxes first 2 blue and 7 red balls second
4 blue and 3 red balls - Bob selects a ball by first choosing one of the
two boxes, and then one ball from this box. - If Bob has selected a red ball, what is the
probability that he selected a ball from the
first box. - An event E Bob has chosen a red ball.
- An event F Bob has chosen a ball from the first
box. - We want to find p(F E)
57Example 2
- Suppose 1 of population has AIDS
- Prob. that the positive result is right 95
- Prob. that the negative result is right 90
- What is the probability that someone who has the
positive result is actually an AIDS patient? - H event that a person has AIDS
- D event of positive result
- PDH 0.95 PD ?H 1- 0.9
- PD PDHPHPD?HP?H
- 0.950.010.10.990.1085
- PHD 0.950.01/0.10850.0876
58Whats behind door number three?
- The Monty Hall problem paradox
- Consider a game show where a prize (a car) is
behind one of three doors - The other two doors do not have prizes (goats
instead) - After picking one of the doors, the host (Monty
Hall) opens a different door to show you that the
door he opened is not the prize - Do you change your decision?
- Your initial probability to win (i.e. pick the
right door) is 1/3 - What is your chance of winning if you change your
choice after Monty opens a wrong door? - After Monty opens a wrong door, if you change
your choice, your chance of winning is 2/3 - Thus, your chance of winning doubles if you
change - Huh?
59(No Transcript)
60Monty Hall Problem
Ci - The car is behind Door i, for i equal to 1,
2 or 3. Hij - The host opens Door j after the
player has picked Door i, for i and j equal to
1, 2 or 3. Without loss of generality, assume,
by re-numbering the doors if necessary, that the
player picks Door 1, and that the host then
opens Door 3, revealing a goat. In other words,
the host makes proposition H13 true. Then the
posterior probability of winning by not switching
doors is P(C1H13).
61P(H13 C1 ) 0.5, since the host will always
open a door that has no car behind it, chosen
from among the two not picked by the player
(which are 2 and 3 here)
62The probability of winning by switching is
P(C2H13), since under our assumption switching
means switching the selection to Door 2, since
P(C3H13) 0 (the host will never open the door
with the car)
The posterior probability of winning by not
switching doors is P(C1H13) 1/3.
63Exercises 6, p. 424, and 16, p. 425
64(No Transcript)
65(No Transcript)
66(No Transcript)
67Continuous random variable
68Continuous Prob. Density Function
- 1. Mathematical Formula
- 2. Shows All Values, x, and Frequencies, f(x)
- f(x) Is Not Probability
- 3. Properties
(Value, Frequency)
f(x)
?
f
x
dx
(
)
?
1
x
a
b
All x
(Area Under Curve)
Value
f
x
(
)
a
x
b
?
?
?
0,
69Continuous Random Variable Probability
d
?
P
c
x
d
f
x
dx
(
)
(
)
?
?
?
c
f(x)
Probability Is Area Under Curve!
X
c
d
70Probability mass function
In probability theory, a probability mass
function (pmf) is a function that gives the
probability that a discrete random variable is
exactly equal to some value. A pmf differs from
a probability density function (pdf) in that the
values of a pdf, defined only for continuous
random variables, are not probabilities as such.
Instead, the integral of a pdf over a range of
possible values (a, b gives the probability of
the random variable falling within that range.
Example graphs of a pmfs. All the values of a pmf
must be non-negative and sum up to 1. (right)
The pmf of a fair die. (All the numbers on the
die have an equal chance of appearing on top
when the die is rolled.)
71Suppose that X is a discrete random variable,
taking values on some countable sample space S
? R. Then the probability mass function fX(x)
for X is given by thus Note that this
explicitly defines fX(x) for all real numbers,
including all values in R that X could never
take indeed, it assigns such values a
probability of zero. Example. Suppose that X is
the outcome of a single coin toss, assigning 0
to tails and 1 to heads. The probability that X
x is 0.5 on the state space 0, 1 (this is a
Bernoulli random variable), and hence the
probability mass function is
72Uniform Distribution
- 1. Equally Likely Outcomes
- 2. Probability Density
- 3. Mean Standard Deviation
f(x)
x
d
c
Mean Median
73Uniform Distribution Example
- Youre production manager of a soft drink
bottling company. You believe that when a
machine is set to dispense 12 oz., it really
dispenses 11.5 to 12.5 oz. inclusive. - Suppose the amount dispensed has a uniform
distribution. - What is the probability that less than 11.8 oz.
is dispensed?
74Uniform Distribution Solution
f(x)
1.0
x
11.5
12.5
11.8
- P(11.5 ? x ? 11.8) (Base)(Height)
- (11.8 - 11.5)(1) 0.30
75Normal Distribution
- 1. Describes Many Random Processes or Continuous
Phenomena - 2. Can Be Used to Approximate Discrete
Probability Distributions - Example Binomial
- Basis for Classical Statistical Inference
- A.k.a. Gaussian distribution
76Normal Distribution
- 1. Bell-Shaped Symmetrical
- 2. Mean, Median, Mode Are Equal
- 4. Random Variable Has Infinite Range
Mean
light-tailed distribution
77Probability Density Function
- f(x) Frequency of Random Variable x
- ? Population Standard Deviation
- ? 3.14159 e 2.71828
- x Value of Random Variable (-?lt x lt ?)
- ? Population Mean
78Effect of Varying Parameters (? ?)
79Normal Distribution Probability
Probability is area under curve!
80Infinite Number of Tables
Normal distributions differ by mean standard
deviation.
Each distribution would require its own table.
Thats an infinite number!
81Standardize theNormal Distribution
Normal Distribution
Standardized Normal Distribution
One table!
82Intuitions on Standardizing
- Subtracting ? from each value X just moves the
curve around, so values are centered on 0 instead
of on ? - Once the curve is centered, dividing each value
by ?gt1 moves all values toward 0, pressing the
curve
83Standardizing Example
Normal Distribution
84Standardizing Example
Normal Distribution
Standardized Normal Distribution
85(No Transcript)
866.4 Expected Value and VarianceLongin Jan
LateckiTemple University
- Slides for a Course Based on the TextDiscrete
Mathematics Its Applications (6th Edition)
Kenneth H. Rosen based on slides by Michael P.
Frank
87Expected Values
- For any random variable V having a numeric
domain, its expectation value or expected value
or weighted average value or (arithmetic) mean
value ExV, under the probability distribution
Prv p(v), is defined as - The term expected value is very widely used for
this. - But this term is somewhat misleading, since the
expected value might itself be totally
unexpected, or even impossible! - E.g., if p(0)0.5 p(2)0.5, then ExV1, even
though p(1)0 and so we know that V?1! - Or, if p(0)0.5 p(1)0.5, then ExV0.5 even
if V is an integer variable!
88Derived Random Variables
- Let S be a sample space over values of a random
variable V (representing possible outcomes). - Then, any function f over S can also be
considered to be a random variable (whose actual
value f(V) is derived from the actual value of
V). - If the range R rangef of f is numeric, then
the mean value Exf of f can still be defined,
as
89Recall that a random variable X is actually a
function f S ? X(S), where S is the sample
space and X(S) is the range of X. This fact
implies that the expected value of X is
Example 1. Expected Value of a Die. Let X be the
number that comes up when a die is rolled.
90Example 2
- A fair coin is flipped 3 times. Let S be the
sample space of 8 possible outcomes, and let X be
a random variable that assignees to an outcome
the number of heads in this outcome. - E(X) 1/8X(TTT) X(TTH) X(THH) X(HTT)
X(HHT) X(HHH) X(THT) X(HTH) 1/80 1
2 1 2 3 1 2 12/8 3/2
91Linearity of Expectation Values
- Let X1, X2 be any two random variables derived
from the same sample space S, and subject to the
same underlying distribution. - Then we have the following theorems
- ExX1X2 ExX1 ExX2
- ExaX1 b aExX1 b
- You should be able to easily prove these for
yourself at home.
92Variance Standard Deviation
- The variance VarX s2(X) of a random variable
X is the expected value of the square of the
difference between the value of X and its
expectation value ExX - The standard deviation or root-mean-square (RMS)
difference of X is s(X) VarX1/2.
93Example 15
- What is the variance of the random variable X,
where X is the number that comes up when a die is
rolled? - V(X) E(X2) E(X)2
- E(X2) 1/612 22 32 42 52 62 91/6
- V(X) 91/6 (7/2) 2 35/12 2.92