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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

According to a study done by Nick Wilson of Otago University Wellington, the probability a randomly selected individual will cover his or her mouth with a tissue, handkerchief, or elbow (the method recommended by public health officials) when sneezing is \(0.047 .\) Suppose you sit on a bench in a mall and observe people's habits as they sneeze. (a) What is the probability that among 15 randomly observed sneezing individuals exactly 2 cover their mouth with a tissue, handkerchief, or elbow? (b) What is the probability that among 15 randomly observed sneezing individuals fewer than 3 cover their mouth with a tissue, handkerchief, or elbow? (c) Would you be surprised if, after observing 15 sneezing individuals, more than 4 covered the mouth with a tissue, handkerchief, or elbow?

(a) construct a discrete probability distribution for the random variable \(X\) [Hint: \(\left.P\left(x_{i}\right)=\frac{f_{i}}{N}\right]\), (b) draw a graph of the probability distribution, (c) compute and interpret the mean of the random variable \(X,\) and \((d)\) compute the standard deviation of the random variable \(X\). $$ \begin{array}{cc} x \text { (games played) } & \text { Frequency } \\ \hline 4 & 18 \\ \hline 5 & 18 \\ \hline 6 & 20 \\ \hline 7 & 35 \end{array} $$

State the criteria for a binomial probability experiment.

In the following probability distribution, the random variable \(X\) represents the number of marriages an individual aged 15 years or older has been involved in. $$ \begin{array}{ll} x & P(x) \\ \hline 0 & 0.272 \\ \hline 1 & 0.575 \\ \hline 2 & 0.121 \\ \hline 3 & 0.027 \\ \hline 4 & 0.004 \\ \hline 5 & 0.001 \end{array} $$ (a) Verify that this is a discrete probability distribution. (b) Draw a graph of the probability distribution. Describe the shape of the distribution. (c) Compute and interpret the mean of the random variable \(X\). (d) Compute the standard deviation of the random variable \(X\). (e) What is the probability that a randomly selected individual 15 years or older was involved in two marriages? (f) What is the probability that a randomly selected individual 15 years or older was involved in at least two marriages?

Explain what 鈥渟uccess鈥� means in a binomial probability experiment.
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