BA Chapter 5a (pages 126-142)

Question 1

Probability, Experiment, and Outcome

Answer 1

Probability is the likelihood that an outcome occurs.

An experiment is the process that results in an outcome.

The outcome of an experiment is a result that we observe.

Question 2

Empirical Probability Distribution (also known as relative frequency)

Answer 2

Use sample data to approximate a probability distribution.

Question 3

Bernoulli Distribution

Answer 3

The Bernoulli distribution is a discrete distribution having two possible outcomes labelled by n = 0 and n = 1 in which n = 1 (“success”) occurs with probability p and n = 0 (“failure”) occurs with probability q = 1 - p.

Question 4

Binomial Distribution

Answer 4

n independent replications of a Bernoulli trial with

• Only two possible outcomes: “success” = p; “failure” = (1 - p) = q
• Constant probabilities on each trial
• Fixed number of trials (n)
• Independent trials

and x = number of successes

Question 5

Complement of an Event

Answer 5

Complement = Opposite

A^c, is the complement (opposite) of A. It consists of all outcomes in the sample space that are not in A. A^c is also written as A′.

A^c = {not A}

P(A^c) = 1 − P(A)

Question 6

Marginal, Joint, & Conditional Probability

Answer 6

• Marginal probability – probability associated with a single outcome of a random variable (a simple, single event).
  
  • P(A)
• Joint probability – probability of outcomes of 2 or more random variables.
  • P(A and B)
• Conditional probability – the probability of the occurrence of A, given that B has already occurred.
  
  • P (A I B) - it is the simplified form of Bayes Theorem

Question 7

• A Random Variable is a numerical description of the outcome of an experiment.
• 2 Types of Random Variables: Discrete and Continuous Random Variable

Answer 7

A discrete random variable is one for which the number of possible outcomes can be counted.

• outcomes of dice rolls
• whether a customer likes or dislikes a product
• number of hits on a Web site link today

A continuous random variable has outcomes over one or more continuous intervals of real numbers.

• weekly change in DJIA
• daily temperature
• time between machine failures

Question 8

Cumulative Distribution Function

Answer 8

F(x) = P(X ≤ x)

• ≤ Sums all the probabilities from x and below, including the probability for x.
• < Sums all the probabilities below x, not including the probability for x.

Question 9

• Sample Space
• Event
• Independent Events
• Mutually Exclusive Events

Answer 9

• Sample space is the collection of all possible outcomes of an experiment. For example, roll a single dice the outcomes are 1,2,3,4,5,6.
• An event is a collection of one of outcomes from a sample space.
• Independent Events Example: Rolling pairs of dice are independent events since they do not depend on the previous rolls.
• 2 events are mutually exclusive if only one event can occur at a time and never both at the same time.

Question 10

Expected Value

Answer 10

The mean of a discrete distribution is referred to as the Expected Value or E(X). It is a weighted average where the weights are the probabilities. The Expected Value of a discrete random variable is the probability-weighted average of all possible values.

Question 11

Exponential Distribution

Answer 11

Exponential Distribution is the probability distribution that describes the time between randomly occurring events, i.e. a process in which events occur continuously and independently at a constant average rate.

Question 12

Goodness of Fit

Answer 12

• The basis for fitting data to a probability distribution.
• Attempts to draw conclusions about the nature of the distribution.
• 3 statistics measure goodness of fit:
  
  • Chi-square
  • Kolmogorov-Smirnov
  • Anderson-Darlin

Question 13

Addition and Multiplication Rule of Probability

Answer 13

• Addition Rule 1
  
  • When two events, A and B, are mutually exclusive, the probability that A or B will occur is the sum of the probability of each event:

P(A or B) = P(A) + P(B)

• Addition Rule 2
  
  • When two events, A and B, are non-mutually exclusive, the probability that A or B will occur is:

P(A or B) = P(A) + P(B) - P(A and B)

• Multiplication Rule for Independent Events
  
  • A method for finding the probability that both 2 events occur.
  • If 2 events are independent, then:

P (A and B) = P(A)*P(B)

Question 14

Normal Distribution

Answer 14

• f(x) is a bell-shaped curve
• Characterized by 2 parameters
  • u (mean, location parameter)
  • σ² (variance, scale parameter)
• The line down the middle is the mean u
• The X-axis is measured in terms of the number of standard deviations σ from the mean u.
• Other Properties:
  
  • Symmetric about the mean (Mean = Median = Mode)
  • Unbounded (goes to infinity on both ends)
  • Empirical rules apply: Empirical Rule means 68% of data fall within u ± 1σ, 95% fall within u ± 2σ, and 99.7% fall within 3σ of the mean.

Question 15

Standard Normal Distribution

Answer 15

• Z is the standard normal random variable with:
  
  • u = 0
  • σ = 1

Question 16

Poisson Distribution

Answer 16

• Models the number of occurrences in some unit of measure (often time or distance).
• There is no limit on the number of occurrences and they are independent of each other.
• This distribution can take on any non-negative integer value.
• Poisson distribution approximates the binomial distribution when n is large and p is small.
• E[X]= λ (lamda)

Question 17

Probability Density Function

Answer 17

• A curve described by a mathematical function that characterizes a continuous random variable. A probability density function (PDF) is a function that describes the relative likelihood for this continuous variable to take on a given value.
• Properties of a probability density function:
  
  • f(x) ≥ 0 for all values of x
  • Total area under the density function equals 1
  • P(X = x) = 0
  • Probabilities are only defined over an interval
  • P(a ≤ X ≤ b) is the area under the density function between a and b.

Question 18

• Random Numbers
• Random number seed
• Random variable

Answer 18

• Random numbers are the basis for generating random samples from probability distributions. Random numbers are uniformly distributed on the interval from 0 to 1.
• Random number seed is a value from which a stream of random numbers is generated. By specifying the same seed, you can produce the same random numbers at a later time.
• A random variable is a numerical description of the outcome of an experiment. Assign real numbers to the outcomes.

Question 19

Uniform Distribution

Answer 19

• All outcomes between a minimum (a) and a maximum (b) are equally likely.
• Useful for generating random numbers.

Question 20

Union 并集

Answer 20

The union of 2 sets is the set of elements that belong to one or both sets. Symbolically, the union of A and B is denoted by A ∪ B.

Example, A = {1, 2, 3}, B = {1, 3, 5}. Then, the union of sets A and B would be {1, 2, 3, 5}.

NOTE:

P(A or B) = UNION of events A and B.

If A and B are mutually exclusive, P(A or B) = P(A)+P(B)

If A and B are not mutually exclusive, P(A orB) = P(A)+P(B)-P(A and B)