Elementary Statistics

1
Elementary Statistics

• Chapter 7

2
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.

3
Lets Do It!

• Pg. 338, 7.5

4
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.

5
Lets Do It!

• Pg. 390, 7.7

6
Set Notation

• Union A or B
• Intersection A and B
• Complement not A

7
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.

8
Basic Probability Rules

• 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. 0P(A)1.
• 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. The probability that an event occurs is 1
  minus the probability that the event does not
  occur. P(A)1-P(Ac).

9
Lets Do It!

• Pg. 396, 7.11

10
The Addition Rule

• 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)

11
The Addition Rule

• 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)

12
Conditional Probability

• The Conditional probability of the event A
  occurring, given that event B has occurred, is
  given by

13
Conditional Probability

• We could rewrite this rule and have an expression
  for calculating an intersection, called the
  multiplication rule.

14
Independent Events

• Two events A and B are independent if P(AB)
  P(A)
• If two events A and B are independent, then

15
Lets Do It!

• Pg. 404, 7.16

16
Lets Do It!

• Pg. 413, 7.17

17
Lets Do It!

• Pg. 419, 7.19

18
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.

19
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.

20
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.

21
Lets Do It!

• Pg. 426, 7.20

22
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

23
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

24
Standard Deviation of X

• And the standard deviation of X is given by

25
A Particular Discrete Distribution, The Binomial
Distribution

• A population with a binomial distribution is a
  discrete population with a particular set of
  assumptions.

26
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

27
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

28
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
  q1-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

29
Binomial Distribution

• Mean

• Variance

• Standard deviation

30
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.

31
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.