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Chapter 6: Problem 12

Determine whether the distribution is a discrete probability distribution. If not, state why. $$ \begin{array}{cc} x & P(x) \\ \hline 1 & 0 \\ \hline 2 & 0 \\ \hline 3 & 0 \\ \hline 4 & 0 \\ \hline 5 & 1 \\ \hline \end{array} $$

Short Answer

Expert verified
Yes, it is a discrete probability distribution since the probabilities sum up to 1 and are all non-negative.

Step by step solution

01

Understand the Problem

The task is to determine if the table represents a discrete probability distribution. A discrete probability distribution lists each possible value the random variable can take along with its probability.
02

Verify that Probabilities Sum to 1

The sum of all probabilities in a discrete probability distribution must be 1. Here, we sum the given probabilities:\[ P(1) + P(2) + P(3) + P(4) + P(5) = 0 + 0 + 0 + 0 + 1 = 1 \]
03

Check Non-negativity of Probabilities

In a discrete probability distribution, each probability must be between 0 and 1 inclusive. Here, all given probabilities (0, 0, 0, 0, 1) meet this requirement.
04

Conclusion

Since the sum of the probabilities is 1 and all probabilities are between 0 and 1, the given table represents a discrete probability distribution.

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Key Concepts

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

Probability
Probability is a measure of how likely an event is to occur. It is a foundational concept in statistics and is often expressed as a number between 0 and 1. A probability of 0 means the event will not happen, and a probability of 1 means the event will definitely happen. For example, if you toss a fair coin, the probability of getting heads is 0.5. In a discrete probability distribution, each outcome of a random variable has a corresponding probability value.
  • Probability measures the chance of occurrence
  • Expressed between 0 and 1
  • Can be applied to simple and complex events
To determine if a distribution is a discrete probability distribution, it is important to check that the sum of all probabilities equals 1 and that each probability is non-negative.
Random Variable
A random variable is a variable whose possible values are numerical outcomes of a random phenomenon. In the context of discrete probability distributions, a random variable can take on a finite or countably infinite set of values. Each value of the random variable has an associated probability. For example, consider a dice roll. The random variable might be the value that comes up on the die, which can be 1, 2, 3, 4, 5, or 6.
  • Represents outcomes of a random phenomenon
  • Can be discrete or continuous
  • Has associated probabilities
In the provided exercise, the random variable takes values 1, 2, 3, 4, and 5. Each of these values has a corresponding probability as given in the table.
Non-negativity
Non-negativity is a crucial requirement for all probabilities in a discrete probability distribution. This means that each probability value must be greater than or equal to 0. Probabilities cannot be negative because they represent the likelihood of an event occurring. Negative probabilities would be nonsensical and impossible in this context.
  • Probabilities must be 鈮� 0
  • Ensures meaningful measurements of likelihood
In the given distribution table, each probability (0, 0, 0, 0, 1) meets the non-negativity requirement, indicating that it could be part of a valid discrete probability distribution.
Probability Sum
The total of all probabilities in a discrete probability distribution must sum to 1. This rule ensures that the distribution covers all possible outcomes of the random variable. If the probabilities do not add up to 1, the distribution is invalid because it signifies that some probability is unaccounted for or over-represented.
  • Sum of probabilities must be 1
  • Guarantees all possible outcomes are covered
In the provided step-by-step solution, the probabilities are summed as follows: \[ P(1) + P(2) + P(3) + P(4) + P(5) = 0 + 0 + 0 + 0 + 1 = 1 \] Since the sum is 1, this indicates that the given probabilities form a valid discrete probability distribution.

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Most popular questions from this chapter

A probability distribution for the random variable \(X,\) the number of trials until a success is observed, is called the geometric probability distribution. It has the same criteria as the binomial distribution (see page 328 ), except that the number of trials is not fixed. Its probability distribution function (pdf) is $$ P(x)=p(1-p)^{x-1}, \quad x=1,2,3, \ldots $$ where \(p\) is the probability of success. (a) What is the probability that Shaquille O'Neal misses his first two free throws and makes the third? Over his career, he made \(52.4 \%\) of his free throws. That is, find \(P(3)\) (b) Construct a probability distribution for the random variable \(X,\) the number of free-throw attempts of Shaquille O'Neal until he makes a free throw. Construct the distribution for \(x=1,2,3, \ldots, 10 .\) The probabilities are small for \(x>10\) (c) Compute the mean of the distribution, using the formula presented in Section 6.1 (d) Compare the mean obtained in part (c) with the value \(\frac{1}{p}\). Conclude that the mean of a geometric probability distribution is \(\mu_{X}=\frac{1}{p} .\) How many free throws do we expect Shaq to take before we observe a made free throw?

What are the two requirements for a discrete probability distribution?

BlackJack is a popular casino game in which a player is dealt two cards where the value of the card corresponds to the number on the card, face cards are worth ten, and aces are worth either one or eleven. The object is to get as close to 21 as possible without going over and have cards whose value exceeds that of the dealer. A blackjack is an ace and a ten in two cards. It pays 1.5 times the bet. The dealer plays last and must draw a card with sixteen and hold with seventeen or more. The following distribution shows the winnings and probability for a \(\$ 20\) bet. In cases where the dealer and player have the same value, there is a tie (called a "push"). Source: "Examining a Gambler's Claims: Probabilistic Fact-Checking and Don Johnson's Extraordinary Winning Streak" by W.J. Hurley, Jack Brimberg, and Richard Kohar. Chance Vol. 27.1,2014 $$ \begin{array}{cc} \text { Winnings } & \text { Probability } \\ \hline 0 & 0.0982 \\ \hline \$ 30 & 0.0483 \\ \hline \$ 20 & 0.389275 \\ \hline-\$ 20 & 0.464225 \end{array} $$ (a) Compute and interpret the expected value of the game from the player's point of view. (b) Suppose over the course of one hour, a player can expect to be dealt about 40 hands. How much should a player expect to win or lose over the course of three hours?

In your own words, provide an interpretation of the mean (or expected value) of a discrete random variable.

Clarinex-D is a medication whose purpose is to reduce the symptoms associated with a variety of allergies. In clinical trials of Clarinex-D, \(5 \%\) of the patients in the study experienced insomnia as a side effect. (a) If 240 users of Clarinex-D are randomly selected, how many would we expect to experience insomnia as a side effect? (b) Would it be unusual to observe 20 patients experiencing insomnia as a side effect in 240 trials of the probability experiment? Why?
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