Session 4

A numerical measure of the likelihood that an event will occur.

A process that generates well-defined outcomes.

The set of all experimental outcomes

An element of the sample space. A sample point represents an experimental outcome

A graphical representation that helps in visualizing a multiple-step experiment

A method of assigning probabilities that is appropriate when all the experimental outcomes are equally likely.

A method of assigning probabilities that is appropriate when data are available to estimate the proportion of the time the experimental outcome will occur if the experiment is repeated a large number of times.

A method of assigning probabilities on the basis of judgement.

A numerical description of the outcome of an experiment.

A random variable that may assume either a finite number of values or an infinite sequence of values.

A random variable that may assume any numerical value in an interval or collection of intervals.

A description of how the probabilities are distributed over the values of the random variable.

A function, denoted by f(x), that provides the probability that x assumes a particular value for a discrete random variable.

A probability distribution for which each possible value of the random variable has the same probability.

A measure of the central location of random variable.

A measure of the variability, or dispersion, of a random variable.

The positive square root of the variance.

A probability distribution showing the probability of x occurrences of an event over a specified interval of time or space.

The function used to compute Poisson probabilities.

For data skewed to the left, the skewness is negative; for data skewed to the right, the skewness is positive. If the data are symmetric, the skewness is zero.

A symmetric distribution, the mean and the median are equal. When the data are positively skewed, the mean will usually be greater than the median; when the data are negatively skewed, the mean will usually be less than the median.

The median provides the preferred measure of location when the data are highly skewed.

1. Smallest value
2. First quartile
3. Median
4. Third quartile
5. Largest Value

1. A box is drawn with the ends of the box located at the first and third quartiles. This box contains the middle 50% of the data.
2. A vertical line is drawn in the box at the location of the median

An advantage of the exploratory data analysis procedure is that they are easy to use; few numerical calculations are necessary. We simply sort the data values into ascending order and identify the five-number summary.

Always assigned on a scale from 0 to 1. A probability near zero indicates an event is unlikely to occur; a probability near 1 indicates an event is almost certain to occur.

• (A) Toss a Coin (B) Head, Tail
• (A) Select a part for inspection (B) Defective, non-defective
• (A) Conduct a sales call (B) Purchase, no purchase
• (A) Roll a die (B) 1, 2, 3, 4, 5, 6
• (A) Play a football game (B) Win, lose, tie

The sample space for an experiment is the set of all experimental outcomes. Also known as sample point.

• S = {Head, Tail}
• S = {Defective, Non-Defective}
  
• S = {1, 2, 3, 4, 5, 6}
  

S = {(H,H), (H,T), (T,H), (T,T)}  Thus, we can see that four experimental outcomes are possible.

If an experiment can be described as a sequence of k steps with n1 possible outcomes on the first step, n2 possible outcomes on the second step, and so on, then the total number of experimental outcomes is given by (n1) (n2) ... (nk)

There are two possible outcomes for the coin toss meaning that (n1=2), (n2=2), etc... Therefore for six coin tosses we will (2)(2)(2)(2)(2)(2) = 64.

1. The probability assigned to each experimental outcome must be between 0 and 1, inclusively. If we let Ei denote the ith experimental outcome and P(Ei) its probability, then this requirement can be written as ---- 0 <= P(Ei) <= 1 for all i ---

When all the experimental outcomes are equally likely. If n experimental outcomes are possible, a probability of 1/n is assigned to each experimental outcome.

P(1) = 1/6, P(2) = 1/6, P(3) = 1/6, P(4) = 1/6, P(5) = 1/6 and P(6) = 1/6

When data are available to estimate the proportion of the time the experimental outcome will occur if the experiment is repeated a large number of times.

When one cannot realistically assume that the experimental outcomes are equally likely and when little relevant data are available. When the subjective method is used to assign probabilities to the experimental outcomes,

we may use any information available, such as our experience or intuition. After considering all available information, a probability value that expresses our degree of belief (on a scale from 0 to 1) that the experimental outcome will occur is specified.

In the physical sciences, researches usually conduct an experiment in a laboratory or a controlled environment in order to learn about cause and effect. In statistical experiments, probability determines outcomes.

Even though the experiment is repeated in exactly he same way, an entirely different outcome may occur. Because of this influence of probability on the outcome, the experiments of statistics are sometimes called random experiments.

The counting rule for combinations is sued to find the number of different samples of size n that can be selected.

An event is a collection of sample points.

The probability of any event is equal to the sum of the probabilities of the sample points in the event.

The sample space, S, is an event. Because it contains all the experimental outcomes, it has a probability of 1; that is, P(S) = 1.

That the experimental outcomes are equally likely. In such cases, the probability of an event can be computed by counting the number of experimental outcomes in the event and dividing the result by the total number of experimental outcomes.

A random variable is a numerical description of the outcome of an experiment.

• (A) Contact five customers (B) Number of customers who place an order (C) 0, 1, 2, 3, 4, 5
• (A) Sell an automobile (B) Gender of the customer (C) 0 if male; 1 if female.

Experimental outcomes based on measurement scales such as time, weight, distance, and temperature can be described by continuous random variables.

• (A) Operate a bank (B) Time between customer arrivals in minutes (C) x => 0
• (A) Construct a new library (B) Percentage of project complete after six months (C) 0 <= x <= 100

Think of the values of the random variable as points on a line segment. Choose two points representing values of the random variable.

Once the probability distribution is known, it is relatively easy to determine the probability of a variety of events that may be of interest to a decision maker.

f(x) = 1/n

where...

n = the number of values the random variable may have

Evaluating f(x) for a given value of the random variable will provide the associated probability. For example, using the preceding probability function, we see that f(2) = 2/10 provides the probability that the random variable assumes a value of 2.

E(x) = μ = ∑xf(x)

To compute the expected value of a discrete random variable, we must multiply each value of the random variable by the corresponding probability f(x) and then add the resulting products.

Var(x) = σ²= ∑(x-μ)f²(x)

An essential part of the variance formula is the deviation, x-μ, which measures how far a particular value of the random variable is from the expected value, or mean, μ.

In computing the variance of a random variable, the deviations squared and then weighted by the corresponding value of the probability function. The sum of these weighted squared deviations for all values of the random variable is called variance.

The notations Var(x) and σ² are both used to denote the variance of a random variable.

The standard deviation is measured in the same units as the random variable and therefore is often preferred in describing the variability of a random variable. The variance σ² is measured in squared units and is thus more difficult to interpret.

1. The probability of an occurrence is the same for any two intervals of equal length.
2. The occurrence or nonoccurrence in any interval is independent of the occurrence or nonoccurrence in any other interval.

f(x) = (μ² e^{-^}μ) - x!

Where...

f(x) = the probability of x occurrences in an interval

μ = Expected value or mean number of occurrences in an interval

e = 2.71828

When computing a Poisson probability for a different time interval, we must first convert the mean arrival rate to the time period of interest and then computer the probability.