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Chapter 5: Problem 7

Briefly explain the two characteristics (conditions) of the probability distribution of a discrete random variable.

Short Answer

Expert verified
The two characteristics or conditions of a probability distribution of a discrete random variable are: 1. Non-negative probabilities i.e., for any outcome 'i' for the variable, the probability \( P(X_i) \) is 0 or a positive number. 2. The sum of probabilities for all outcomes is equal to 1 i.e., \( \sum_{i} P(X_i) = 1 \), where 'i' represents all possible outcomes.

Step by step solution

01

Identification of Characteristics

There are two fundamental characteristics or conditions of the probability distribution for a discrete random variable: 1. Non-Negative Probabilities: The probability of each outcome for discrete random variable is non-negative. This means, for each individual outcome 'i', denoted by \( X_i \), the probability \( P(X_i) \) should be 0 or a positive number.2. Sum of Probabilities: The sum of probabilities of all possible outcomes for the discretely random variable should be equal to 1. Expressed mathematically, \( \sum_{i} P(X_i) = 1 \), where 'i' represents all possible outcomes.
02

Description of First Characteristic

Discussing the first characteristic, non-negative probabilities, this ensures that no outcome has a 'negative chance' of happening. Since probabilities represent how likely an event is, it makes no logical sense for an event to have a less than zero probability. Therefore, this rule ensures the validity and coherence of the probability model. This also implies that a discrete random variable's probability distribution will always have probabilities that are 0 or higher.
03

Description of Second Characteristic

Moving onto the second characteristic, the sum of probabilities, it is also a logical necessity for valid probability assignment. This rule means that when you add up all the different probabilities for all the possible outcomes of a random variable, you get 1. Essentially, this rule says that some event from the set of possible events has to happen. As a consequence, the sum of the probabilities of all possible outcomes in the probability distribution is always equal to 1.

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Most popular questions from this chapter

A history teacher has given her class a list of seven essay questions to study before the next test. The teacher announced that she will choose four of the seven questions to give on the test, and each student will have to answer three of those four questions. a. In how many ways can the teacher choose four questions from the set of seven? b. Suppose that a student has enough time to study only five questions. In how many ways can the teacher choose four questions from the set of seven so that the four selected questions include both questions that the student did not study? c. What is the probability that the student in part b will have to answer a question that he or she did not study? That is, what is the probability that the four questions on the test will include both questions that the student did not study?

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