/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 7 Briefly explain the two characte... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

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.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The following table gives the probability distribution of the number of camcorders sold on a given day at an electronics store. $$ \begin{array}{l|ccccccc} \hline \text { Camcorders sold } & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline \text { Probability } & .05 & .12 & .19 & .30 & .20 & .10 & .04 \\ \hline \end{array} $$ Calculate the mean and standard deviation for this probability distribution. Give a brief interpretation of The value of the mean.

A university police department receives an average of \(3.7\) reports per week of lost student ID cards. a. Find the probability that at most 1 such report will be received during a given week by this police department. Use the Poisson probability distribution formula. b. Using the Poisson probabilities table, find the probability that during a given week the number of such reports received by this police department is i. 1 to 4 ii. at least 6 iii. at most 3

Suppose the owner of a salvage company is considering raising a sunken ship. If successful, the venture will yield a net profit of \(\$ 10\) million. Otherwise, the owner will lose \(\$ 4\) million. Let \(p\) denote the probability of success for this venture. Assume the owner is willing to take the risk to go ahead with this project provided the expected net profit is at least \(\$ 500,000\). a. If \(p=.40\), find the expected net profit. Will the owner be willing to take the risk with this probability of success? b. What is the smallest value of \(p\) for which the owner will take the risk to undertake this project?

Despite all efforts by the quality control department, the fabric made at Benton Corporation always contains a few defects. A certain type of fabric made at this corporation contains an average of \(.5\) defects per 500 yards. a. Using the Poisson formula, find the probability that a given piece of 500 yards of this fabric will contain exactly 1 defect. b. Using the Poisson probabilities table, find the probability that the number of defects in a given 500-yard piece of this fabric will be i. 2 to 4 ii. more than 3 iii. less than 2

In the 2008 Beach to Beacon \(10 \mathrm{~K}\) run, \(27.4 \%\) of the 5248 participants finished the race in \(49 \mathrm{~min}\) utes 42 seconds \((49: 42)\) or faster, which is equivalent to a pace of less than 8 minutes per mile (Source: http://www.beach2beacon.org/b2b_2008_runners.htm). Suppose that this result holds true for all people who would participate in and finish a \(10 \mathrm{~K}\) race. Suppose that two \(10 \mathrm{~K}\) runners are selected at random. Let \(x\) denote the number of runners in these two who would finish a \(10 \mathrm{~K}\) race in \(49: 42\) or less. Construct the probability distribution table of \(x\). Draw a tree diagram for this problem.
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Geometry

Read Explanation

Pure Maths

Read Explanation

Logic and Functions

Read Explanation

Calculus

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.