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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 conditions of the probability distribution of a discrete random variable are: 1) Every outcome is assigned a probability value in the range 0 to 1 (inclusive). 2) The sum of all the probabilities of all the possible outcomes is exactly 1.

Step by step solution

01

Condition 1: Probability Assignment

The first characteristic of a probability distribution of a discrete random variable is that each outcome in the sample space is assigned a probability between 0 and 1. In mathematical terms, for any event E, 0 鈮� P(E) 鈮� 1 where P(E) is the probability of event E.
02

Condition 2: Probability Sum

The second characteristic of a probability distribution is that the sum of all probabilities of the possible outcomes in the sample space is equal to 1. Putting it mathematically, if the sample space is denoted by S, then the sum of probabilities of all events in S is 1, i.e., 鈭� P(E) = 1 for all E 鈭� S.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Discrete Random Variable
A discrete random variable is a type of statistical variable that can take on only distinct, separate values. It is called "discrete" because there is no possibility of values between these distinct points.
For example, consider the number of students in a classroom. This is a discrete random variable because you can have 20 or 21 students, but not 20.5 students.
Discrete random variables are fundamental in probability distributions because they define the basic elements that the probability distribution seeks to characterize.
  • Unlike continuous variables, which can take any value in a range, discrete variables have countable outcomes.
  • The possible values of a discrete random variable could be finite or infinite, like a geometric series or a dice roll.
Understanding discrete random variables helps us assign probabilities to each possible outcome, forming the foundation of probability assignments.
Probability Assignment
Probability assignment involves determining the probability for each potential outcome of a discrete random variable. This is a crucial step in building the probability distribution.
Each outcome or event is assigned a probability value that lies between 0 and 1, inclusive. In mathematical expressions, we represent this as \(0 \leq P(E) \leq 1\).
These assignments are significant as they express the likelihood of occurrence for each possible outcome. For example, in a fair dice roll, each number from 1 to 6 has a probability of \(\frac{1}{6}\).
  • These probabilities help us understand the chance of occurrence of specific events.
  • Correctly assigning probabilities is necessary to ensure the probability sum condition is met.
With these assignments, you can assess risk, plan strategies, and make informed predictions based on available data.
Sample Space
Sample space refers to the complete set of all possible outcomes for a given experiment involving a discrete random variable. It is represented commonly by the symbol \(S\).
Knowing the sample space is essential for defining the probability distribution because it bounds what all the included random outcomes are. For example, in a coin flip, the sample space is \(\{\text{Heads, Tails}\}\).
  • Each outcome in the sample space should be distinct and collectively exhaustive.
  • Listing out a sample space helps visualize all possible outcomes.
Creating an accurate sample space is crucial as it becomes the basis to verify if the sum of probabilities equals 1, ensuring accurate probability distribution formulation.
Probability Sum
The probability sum is the requirement that the total of the probability values assigned to all possible outcomes within a sample space equals 1.
This condition certifies the foundation of a valid probability distribution. Mathematically, for a sample space \(S\), this is expressed as \(\sum P(E) = 1\) for all \(E \in S\).
The concept of probability sum asserts that every possible outcome is considered, and their collective probabilities encapsulate all that can happen in the experiment.
  • Ensures that no probability is neglected or overcounted.
  • Helps validate the constructed probability distribution model.
Following the probability sum rule confirms that the probability model is complete and consistent with the principles of probability, making it reliable for further statistical analysis.

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Most popular questions from this chapter

Let \(x\) be a discrete random variable that possesses a binomial distribution. Using the binomial formula, find the following probabilities. a. \(P(x=5)\) for \(n=8\) and \(p=.70\) b. \(P(x=3)\) for \(n=4\) and \(p=.40\) c. \(P(x=2)\) for \(n=6\) and \(p=.30\) Verify your answers by using Table I of Appendix C.

The number of calls that come into a small mail-order company follows a Poisson distribution. Currently, these calls are serviced by a single operator. The manager knows from past experience that an additional operator will be needed if the rate of calls exceeds 20 per hour. The manager observes that 9 calls came into the mail-order company during a randomly selected 15 -minute period. a. If the rate of calls is actually 20 per hour, what is the probability that 9 or more calls will come in during a given 15 -minute period? b. If the rate of calls is really 30 per hour, what is the probability that 9 or more calls will come in during a given 15 -minute period? c. Based on the calculations in parts a and \(\mathrm{b}\), do you think that the rate of incoming calls is more likely to be 20 or 30 per hour? d. Would you advise the manager to hire a second operator? Explain.

A high school boys' basketball team averages \(1.2\) technical fouls per game. a. Using the appropriate formula, find the probability that in a given basketball game this team will commit exactly 3 technical fouls. b. Let \(x\) denote the number of technical fouls that this team will commit during a given basketball game. Using the appropriate probabilities table from Appendix \(\mathrm{C}\), write the probability distribution of \(x\).

Explain the meaning of a random variable, a discrete random variable, and a continuous random variable. Give one example each of a discrete random variable and a continuous random variable.

What is the parameter of the Poisson probability distribution, and what does it mean?
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