Question

List the characteristics of a multinomial experiment. (Select all that apply.)

The number of successes is evenly distributed over all k categories.

We are interested in x, the number of events that occur in a period of time or space.

The trials are independent.

The experiment consists of n identical trials.

Its mean is 0 and its standard deviation is 1.

The outcome of each trial falls into one of k categories.

The experiment contains M successes and N − M failures.

The probability that the outcome of a single trial falls into a particular category remains constant from trial to trial.

The probability that the outcome of a single trial falls between two categories is equal to the area under the curve between those categories.

We are interested in x, the number of successes observed during the n trials.

The experimenter counts the observed number of outcomes in each category.

Each trial results in one of only two possible outcomes.

Answer 1

Definition of multinomial distribution-

Suppose, are random variables with corresponding probabilities to occur in each trial. For n trials, the probability mass function is given by

Characteristics of multinomial distribution-

• We are interested in x, the number of events that occur in a period of time or space.
• The trials are independent.
• The experiment consists of n identical trials. In each trial, probability of occurring certain event is identical.
• The outcome of each trial falls into one of k categories. All the events are mutually exclusive (disjoint) and exhaustive.
• The probability that the outcome of a single trial falls into a particular category remains constant from trial to trial. The experiment is continued in identical condition.
• The probability that the outcome of a single trial falls between two categories is equal to the area under the curve between those categories. The categories are disjoint and this interval probability is additive.
• The experimenter counts the observed number of outcomes in each category.

Deficiencies of other points mentioned-

• In general, the number of successes is not evenly distributed over all k categories. However, it may happen if .
• In general, its mean is not 0 and its standard deviation is not 1. These values depend on n and 's.
• In general, we can not conclude like 'the experiment contains M successes and N − M failures'. This conclusion can be made in special case called binomial distribution when k=2.
• In general, we are interested in , I=1,2,...,k the number of successes observed in each category during the n trials. The given statement is true in special case of binomial distribution when k=2.
• In general, each trial does not result in one of only two possible outcomes. However, the given statement is true in special case of binomial distribution when k=2.