Question

In Exercises \(\mathrm{P} .74\) to \(\mathrm{P} .77\), fill in the \(?\) to make \(p(x)\) a probability function. If not possible, say so. $$ \begin{array}{lccccc} \hline x & 1 & 2 & 3 & 4 & 5 \\ \hline p(x) & 0.3 & ? & 0.3 & 0.3 & 0.3 \\ \hline \end{array} $$

Short answer

It is not possible to make \(p(x)\) a valid probability function in this case.

Step-by-step solution

Sum Known Probabilities

Start by adding the known probabilities. These are 0.3 for \(p(1)\), 0.3 for \(p(3)\), 0.3 for \(p(4)\), and 0.3 for \(p(5)\). This will give you a total of 1.2.

Determine The Remaining Probability

Since the total probability must equal 1, subtract the sum obtained in Step 1 from 1. This would be \(1 - 1.2\), which reveals a negative value ( -0.2)

Check if Obtained Value is Within Valid Range

Since a probability can not be a negative number, the obtained value in Step 2 ( -0.2) is not valid. Therefore, it is impossible to fill in a probability for \(p(2)\) in such a way that \(p(x)\) becomes a valid probability function.

Key concepts

Probability Theory

Probability theory is the branch of mathematics concerned with analyzing random events and determining the likelihood of various outcomes. It is foundational to understanding how to evaluate risk and uncertainty and is applicable across numerous fields, from finance to physics.

At its core, probability theory deals with the concept of a 'probability', which is a numerical value ranging between 0 and 1, inclusive. This value represents the measure of how likely an event is to occur. A probability of 0 shows that an event is impossible, while a probability of 1 indicates an event is certain to happen. Problems in probability theory often require calculating the probabilities of composite events, based on the likelihood of their individual components.

In the context of the provided textbook exercise, the exercise involves determining if a set of values can form a valid probability function for a discrete random variable. This requires understanding that the sum of probabilities for all possible outcomes of a random variable must equal 1, reflecting certainty that one of the outcomes will occur.

Probability Distribution

A probability distribution is a mathematical function that describes all the possible values and likelihoods that a random variable can take within a given range. Understanding probability distributions is crucial for interpreting and predicting the outcomes of random processes.

There are two main types of probability distributions: discrete and continuous. Discrete probability distributions apply to variables that have distinct, countable outcomes, such as the roll of a die. Continuous probability distributions, on the other hand, apply to variables that can take on a continuous range of values, like heights of people.

The exercise from the textbook is dealing with a discrete random variable, as evidenced by the distinct values that the variable 'x' can take. The task given was to determine the missing probability so that the probabilities add up to 1, adhering to the fundamental property of a probability distribution.

Discrete Random Variable

A discrete random variable is a type of random variable that possesses a countable number of distinct outcomes. Each of these outcomes has an associated probability that quantifies the chance of the outcome occurring. Examples of discrete random variables include the number of heads in a series of coin tosses, or the result of rolling a six-sided die.

In practice, when developing a probability function for a discrete random variable, the probabilities assigned to each possible outcome must sum to 1, ensuring the model is complete and accounts for all possible events. If the sum of the probabilities is less than or more than 1, it indicates an error in the model. As illustrated in the solution to the exercise, the sum of the known probabilities exceeded 1, which demonstrated that it is not possible to assign a valid probability to the remaining outcome, and thereby a valid probability function cannot exist.