# Elementary Statistics

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```					Elementary Statistics
Chapter 7
Sample Space
A sample space is the set of all
possible individual outcomes of a
random process. The sample space is
typically denoted by S and may be
represented as a list, a tree diagram, an
interval of values, a grid of possible
values, and so on.
Let’s Do It!
Pg. 338, 7.5
Event
An event is any subset of the sample
space S. An event A is said to occur if
any one of the outcomes in A occurs
when the random process is performed
once.
Let’s Do It!
Pg. 390, 7.7
Set Notation
Union: A or B
Intersection: A and B
Complement: not A
Disjoint
Two events A and B are disjoint or
mutually exclusive if they have no
outcomes in common. Thus, if one of
the events occurs, the other cannot
occur.
Basic Probability Rules
1. Any probability is always a numerical value
between 0 and 1. The probability is 0 if the
event cannot occur. The probability is 1 if
the event is a sure thing. 0≤P(A)≤1.
2. If we add up the probabilities of each of the
individual outcomes in the sample space,
the total probability must be equal to one;
P(S)=1.
3. 3. The probability that an event occurs is 1
minus the probability that the event does
not occur. P(A)=1-P(Ac).
Let’s Do It!
Pg. 396, 7.11
The probability that either the event A or
the event B occurs is the sum of their
individual probabilities minus the
probability of their intersection.

P(A or B) = P(A) + P(B) - P(A and B)
If the two events A and B do not have any
outcomes in common (disjoint), then the
probability that one or the other occurs
is simply the sum of their individual
probabilities.

P(A or B) = P(A) + P(B)
Conditional Probability
The Conditional probability of the event A
occurring, given that event B has
occurred, is given by

P(A and B)
P(A B) 
P(B)
Conditional Probability
We could rewrite this rule and have an
expression for calculating an
intersection, called the multiplication
rule.
P(A and B)  P(B)P(A B)
Independent Events
Two events A and B are independent if
P(A|B) = P(A)

If two events A and B are independent,
then
P(A and B)  P(B)P(A)
Let’s Do It!
Pg. 404, 7.16
Let’s Do It!
Pg. 413, 7.17
Let’s Do It!
Pg. 419, 7.19
Random Variable
A random variable is an uncertain
numerical quantity whose value
depends on the outcome of a random
experiment. We can think of a random
variable as a rule that assigns one (and
only one) numerical value to each point
of the sample space.
Discrete vs Continuous RVs
A discrete random variable can
assume at most a finite or infinite but
countable number of distinct values. A
continuous random variable can
assume any value in an interval or
collection of intervals.
Probability Distribution
The probability distribution of a discrete
random variable X is a table or rule that
assigns a probability to each of the
possible values of the random variable
X. The values of a discrete probability
distribution must be between 0 and 1
and must add up to 1.
 p 1
i
Let’s Do It!
Pg. 426, 7.20
Expected Value
If X is a discrete random variable taking
on the values x1, x2,…,xk, with
probabilities p1, p2,…,pk, then the mean
or expected value of X is given by

E(X)    x1 p1  x2 p2  ... xk pk  xi pi
Variance of X
If X is a discrete random variable taking
on the values x1, x2,…,xk, with
probabilities p1, p2,…,pk, then the
variance of X is given by
E(X )  
2

           
 E X    x i  pi  E X  EX
2       2        2         2

  x pi 
2       2
i
Standard Deviation of X
And the standard deviation of X is
given by

SD(X)       2
A Particular Discrete
Distribution, The Binomial
Distribution
A population with a binomial distribution
is a discrete population with a particular
set of assumptions.
Combinations
“n choose x” represents the number of
ways of selecting x items (without
replacement) from a set of n
distinguishable items when the order of
the selection is not important and is given
by
n
      n!
, where n! n(n 1)(n  2)...( 2)(1)
x  x!n  x !
Binomial Distribution
A binomial random variable is the total
number of successes in n independent
trials with the following properties:
Each experiment consists of n identical
trials.
Each trial has two possible outcomes
(success/failure).
The trials are independent
Binomial Distribution
A binomial random variable is the total
number of successes in n independent
trials with the following properties:
The probability of success p, remains
the same for each trial. The probability
of a failure is q=1-p.
The binomial random variable X is the
number of successes in the n trials. X =
bin(n,p) and can take on values
0, 1, 2, …, n
Binomial Distribution
Mean:                 E(X)    np

Variance:             VAR(X)    npq
2

Standard deviation:   SD(X)    npq
Continuous Random Variables
The probability distribution of a
continuous random variable X is a
curve such that the area under the
curve over an interval is equal to the
probability that the random variable X is
in the interval. The values of a
continuous probability distribution must
be at least 0 and the total area under
the curve must be 1.
Mean of a Continuous RV
The mean or expected value of a
continuous random variable X is the
point at which the probability density
function would balance.

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 views: 2 posted: 9/29/2012 language: English pages: 31