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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 contractor has submitted bids on three state jobs: an office building, a theater, and a parking garage. State rules do not allow a contractor to be offered more than one of these jobs. If this contractor is awarded any of these jobs, the profits earned from these contracts are $$\$ 10$$ million from the office building, $$\$ 5$$ million from the theater, and $$\$ 2$$ million from the parking garage. His profit is zero if he gets no contract. The contractor estimates that the probabilities of getting the office building contract, the theater contract, the parking garage contract, or nothing are \(.15, .30, .45\), and 10, respectively. Let \(x\) be the random variable that represents the contractor's profits in millions of dollars. Write the probability distribution of \(x\). Find the mean and standard deviation of \(x\). Give a brief interpretation of the values of the mean and standard deviation.

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Nathan Cheboygan, a singing gambler from northern Michigan, is famous for his loaded dice. The following table shows the probability distribution for the sum, denoted by \(x\), of the faces on a pair of Nathan's dice. $$ \begin{array}{l|ccccccccccc} \hline x & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\ \hline P(x) & .065 & .065 & .08 & .095 & .11 & .17 & .11 & .095 & .08 & .065 & .065 \\ \hline \end{array} $$ a. Draw a bar graph for this probability distribution. b. Determine the probability that the sum of the faces on a single roll of Nathan's dice is \(\begin{array}{lll}\text { i. an even number } & \text { ii. } 7 \text { or } 11 & \text { iii. } 4 \text { to } 6\end{array}\) iv. no less than 9

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