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Chapter 4: Problem 2

Determine whether or not the table is a valid probability distribution of a discrete random variable. Explain fully. a. \begin{tabular}{c|ccccc} & 0 & 1 & 1 & 3 & 4 \\ \hline\(P(x)\) & -0.85 & 0.50 & 0.25 & 0.10 & 0.30 \end{tabular}b. \begin{tabular}{c|ccc} & 1 & 1 & 3 \\ \hline\(P(x)\) & 0.325 & 0.406 & 0.164 \end{tabular} C. \begin{tabular}{c|ccccc} & 25 & 26 & 27 & 28 & 20 \\ \hline\(P(x)\) & 0.13 & 0.27 & 0.28 & 0.18 & 0.14 \end{tabular}

Short Answer

Expert verified
Only Table C is a valid probability distribution.

Step by step solution

01

Check Non-Negativity for Table A

Inspect each probability value in the table A. Each value must be non-negative. For Table A: - P(x=0) = -0.85 - P(x=1) = 0.50 - P(x=1) = 0.25 - P(x=3) = 0.10 - P(x=4) = 0.30 P(x=0) = -0.85 is negative, so Table A is not a valid probability distribution.
02

Check Summation for Table A

Sum the probabilities in Table A to see if they equal 1. Sum = -0.85 + 0.50 + 0.25 + 0.10 + 0.30 = 0.30, which is not equal to 1. Hence, Table A is not a valid probability distribution.
03

Check Non-Negativity for Table B

Inspect each probability in table B. Each probability must be non-negative. For Table B: - P(x=1) = 0.325 - P(x=1) = 0.406 - P(x=3) = 0.164 All probabilities are non-negative, so Table B passes the non-negativity check.
04

Check Summation for Table B

Sum the probabilities in Table B to ensure they equal 1. Sum = 0.325 + 0.406 + 0.164 = 0.895, which is not equal to 1. Hence, Table B is not valid.
05

Check Non-Negativity for Table C

Inspect each probability in Table C. Each should be non-negative. For Table C: - P(x=25) = 0.13 - P(x=26) = 0.27 - P(x=27) = 0.28 - P(x=28) = 0.18 - P(x=20) = 0.14 All probabilities are non-negative, so Table C passes the non-negativity check.
06

Check Summation for Table C

Sum the probabilities in Table C to see if they equal 1. Sum = 0.13 + 0.27 + 0.28 + 0.18 + 0.14 = 1.0, which equals to 1. Hence, Table C is a valid probability distribution.

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Key Concepts

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

Discrete Random Variable
When studying probability distributions, understanding discrete random variables is essential. A discrete random variable represents outcomes that can take on only distinct, separate values, often as integers.

For instance, consider a dice roll. The possible outcomes are 1, 2, 3, 4, 5, and 6鈥攅ach number represents a discrete value. In a probability distribution, these outcomes are paired with their likelihood of occurrence, termed probabilities. These probabilities must adhere to specific criteria to form a valid distribution.

Recognizing what constitutes a discrete random variable is foundational for assessing a probability distribution and verifying its validity.
Validity Checking
To determine if a probability distribution is valid, we must perform validity checking. This involves assessing whether the distribution meets two primary conditions: non-negativity and summation.

During this process, you'll evaluate if each probability is non-negative, signifying that every probability should be zero or a positive value. A probability can't be less than zero because negative probabilities lack real-world meaning.

Next, the sum of all probabilities in the distribution must equal exactly one. This sum represents the entirety of possible outcomes for a random variable, confirming that no opportunities for event occurrence are missing.
Non-Negativity and Summation
Non-negativity and summation are key checks for validating probability distributions. These checks ensure that the probability function correctly models real situations.

  • Non-Negativity: Each probability value should be non-negative鈥攝ero or positive. A negative probability is nonsensical, indicating an impossible and undefined event.

  • Summation: The total sum of all individual probability values should be exactly one. This precise totality assures that every possible outcome of the random variable is included in the distribution.

By applying these principles, you verify the distribution logically represents all possible outcomes, confirming its credibility and coherence.
Introductory Statistics
Introductory statistics introduces fundamental concepts crucial for understanding probability distributions, including discrete random variables.

The role of statistics is to provide a framework for collecting, analyzing, and interpreting data. At the introductory level, students learn how to work with distributions that describe the probabilities of different outcomes for random variables.

Through these concepts, learners understand how statistics enable prediction and analysis, forming the basis for more advanced statistical analysis and decision-making in uncertain situations.

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Most popular questions from this chapter

A coin is bent so that the probability that it lands heads up is \(2 / 3\). The coin is tossed ten times. a. Find the probability that it lands heads up at most five times. b. Find the probability that it lands heads up more times than it lands tails up.

Tybalt receives in the mail an offer to enter a national sweepstakes. The prizes and chances of winning are listed in the offer as: \(\$ 5\) million, one chance in 65 million; \(\$ 150,000,\) one chance in 6.5 million; \(\$ 5,000\), one chance in 650,000 ; and \(\$ 1,000,\) one chance in 65,000 . If it costs Tybalt 44 cents to mail his entry, what is the expected value of the sweepstakes to him?

An insurance company will sell a \(\$ 90,000\) one-year term life insurance policy to an individual in a particular risk group for a premium of \(\$ 478\). Find the expected value to the company of a single policy if a person in this risk group has a \(99.62 \%\) chance of surviving one year.

Two fair dice are rolled at once. Let \(X\) denote the difference in the number of dots that appear on the top faces of the two dice. Thus for example if a one and a five are rolled, \(x=4,\) and if two sixes are rolled, \(X=0\). a. Construct the probability distribution for \(X\). b. Compute the mean \(\mu\) of \(X\). c. Compute the standard deviation \(\sigma\) of \(X\).

A fair coin is tossed repeatedly until either it lands heads or a total of five tosses have been made, whichever comes first. Let \(X\) denote the number of tosses made. a. Construct the probability distribution for \(X\). b. Compute the mean \(\mu\) of \(X\). c. Compute the standard deviation \(\sigma\) of \(X\).
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