STATS EXAM #2

CHARACTERISTICS or PROPERTIES OF A PROBABILITY DISTRIBUTION

1- 0 < P(x) < 1 2- (Summation symbol here) P(x) = 1

A RV CAN BE DISCRETE OR CONTINUOUS

1- A DISCRETE RV assumes countable values. The characteristics of a discrete RV are - Finite number or countable values - Consists of the set of whole numbers only: 0, 1, 2, .... , ∞ - Results from counting persons or objects, for example, there are 32 children in the fifth grade. 2- A CONTINUOUS RV assumes uncountable values. The characteristics of a continuous RV are - Infinite number of values - Consists of the set of Rational Numbers > 0 - Results from measuring, for example, the time taken to complete an examination.

• PROPERTIES OF PROBABILITY

1- P(Sample Space) = 1 and P(Impossible Event) = 0 2- Probability of an event 0 < P(A) < 1

DISCRETE PROBABILITY DISTRIBUTION

I- BINOMIAL DISTRIBUTION - The Binomial Probability Distribution is a discrete probability distribution and is applied to experiments that satisfy the following four (4) conditions of a binomial experiment, an experiment that can only have two possible outcomes. Each repetition of a binomial experiment is called a trial or a Bernoulli trial. 1. There are "n" identical trials. Each repetition of a binomial experiment is called a trial. 2. Each trial have two and only two possible outcomes: - The probability of success is denoted by p - The probability of failure is denoted by q; where p + q = 1 and q = 1 - p. 3. The trials are independent (the outcomes of one trial will not affect the probabilities of subsequent trials). As a result of this independence of trials is that 4. The probability of either outcome remains constant during the experiment. Note: A success does not mean that the corresponding outcome is considered favorable or desirable. The outcome to which the question refers is usually called a "success." Binomial Probabilities are found by using the binomial formula, binomial table, computer, calculator, and by using the Normal Distribution as an approximation to the Binomial Distribution. The only values needed to find the probability of "x" successes in n trials are the values of n and p. These, n and p values are called the parameters of the binomial probability distribution or simply the binomial parameters. The Mean and Standard Deviation of a binomial distribution is given by µ = np and Squareroot of PQN = where n = total number of trials p = probability of success q = probability of failure

COMPLEMENTATION RULE - deals with two events

The complement of event "A," denoted by (-A-) , ~A or A' is the event that includes all the outcomes for an experiment that are not in "A." P( -A-) = 1 - P(A) or P(A) = 1 - P( -A-) Note: The two complementary events are always mutually exclusive.

IN A STATISTICAL PROBLEM

a sample is known and we want to make an inference about the population.

OUTCOMES

are the observations obtained from the experiment. For example, when we roll a die the outcomes are 1, 2, 3, 4, 5, and 6. An outcome is what happens.

A PROBABILITY DISTRIBUTION

is the collection of all the possible values that a RV can assume and their corresponding probabilities.

A SAMPLE SPACE

for an experiment consists of all possible simple events or the collection of all outcomes for an experiment. The elements of a sample space are called sample points. The sample space is denoted by "S" and the sample space for rolling a die is S = {1, 2, 3, 4, 5, 6}.

AN EVENT

is a collection of one or more outcomes of an experiment, for example, getting a 6 or an even number (2, 4, 6) when rolling a single die. The event is what you count. Events can be simple or compound

COMPOUND EVENTS

is a combination of simple events (a collection of more than one outcome for an experiment). For example, obtaining an even number (2, 4, 6) or an odd number (1, 3, 5) when we roll a die.

A RANDOM VARIABLE (RV)

is a numerical description of the outcome of a random experiment.

• Probability denoted by "P"

is a numerical measure of the likelihood or chance that a specific event will occur. Capital letters A, B, C, etc. will denote specific events, for example, P(A) = is the Probability that event "A" occurs.

IN A PROBABILITY PROBLEM

the population is known and we want to determine the likelihood, chance or probability of observing a sample.

SUBJECTIVE approach to PROBABILITY

the probabilities are assigned by an individual or group based on whatever evidence is available.

RELATIVE FREQUENCY or EMPIRICAL approach to PROBABILITY

the probabilities are determined using past data or generating new data by performing the experiment large number of times. The relative frequency approach to probability is applied to experiments that do not have equally likely outcomes but can be repeated

CLASSICAL approach to PROBABILITY

the probability that an outcome occurs is determined before the experiment, and refers to events that have equally likely outcomes, that is, outcomes that have the same probability of occurrences.

MULTIPLICATION RULE - The keyword for the multiplication rule is "and" or "both"

a- Independent Events - Two events "A" and "B" are independent if the occurrence of one event does not affect the probability of the occurrence of the other event. A and B events are independent if either P(A| B) = P(A) or P(B| A) = P(B) Sampling with Replacement - The selected element is returned to the population before the next element is selected. Thus, in sampling with replacement, the population contains the same number of elements each time a selection is made. We may select the same item more than once in such a sample. The events are independent, therefore the probabilities remain constant. P(A and B) = P(A) * P(B) b- Dependent Events - Two events "A" and "B" are dependent if the occurrence of one event affects the probability of the occurrence of the other event. Sampling without Replacement. The selected element is not returned to the population for selection. In this case, each time we select an element, the size of the population is reduced by one element. Thus, we cannot select the same element more than once in this type of sampling. The events are dependent therefore the probabilities don't remain constant. P(A and B) = P(A) * P(B| A) Note: P(B| A) is the probability of B given that A has already occurred. Conditional Probability is the probability that an event will occur given that another event has already occurred. If A and B are two events, then the conditional probability of A given B is written as P(A| B) and read as "the probability of A given that B has already occurred."

ADDITION RULE - The keyword for the addition rule is "or"

a. Not Mutually Exclusive Events are events that have common outcomes. The events happen at the same time P(A or B) = P(A) + P(B) - P(A and B) For example, when rolling a die what is the probability of getting an even number or a number less than 5. P(even or <5) = P(2) + P(4) + P(6) + P(1) + P(2) + P(3) + P(4) - [P(2) + P(4)] = 7/6 - 2/6 = 5/6 Joint Probability is the probability of the intersection of two events. It is the likelihood or chance that two or more events will happen at the same time. b. Mutually Exclusive Events are events that have no common outcomes. The events cannot occur together P(A or B) = P(A) + P(B) For example, when rolling a die what is the probability of getting a 2 or a number greater than 5. P(2 or >5) = P(2) + P(6) = 1/6 + 1/6 = 2/6 = 1/3. The joint probability in this case is zero.

AN EXPERIMENT or PROCEDURE

is a process that allows us to obtain observations, and when it is performed, results in one and only one of many observations. For example, rolling a single die. An experiment is what you do.

SIMPLE EVENT OR ELEMENTARY EVENT

is an outcome or an event, which can not be broken down any further; that is, it is a unique representation of an event. The outcome of a four when we roll a die is a simple event, but the outcome of an even number (2, 4, 6), is not a simple event, because it can be broken down into three simple events: 2, 4 or 6.