Question

Use the probability distribution for the random variable \(x\) to answer the questions in Exercises 12-16. $$\begin{array}{l|rrrrrr}x & 0 & 1 & 2 & 3 & 4 & 5 \\\\\hline p(x) & .1 & .3 & .4 & .1 & ? & .05\end{array}$$ Construct a probability histogram to describe \(p(x)\).

Short answer

Answer: The probability for x=4 is 0.05.

Step-by-step solution

Find the missing probability for \(x=4\)

As we know, the sum of all probabilities in a probability distribution must equal 1. Let's calculate the missing probability for \(x=4\). We have the probabilities of other values of \(x\): $$p(0) = 0.1,\ p(1) = 0.3,\ p(2) = 0.4,\ p(3) = 0.1,\ p(5) = 0.05$$ Summing them up and subtracting from 1, we find the missing probability for \(x=4\): $$ p(4) = 1 - (0.1+0.3+0.4+0.1+0.05) = 1 - 0.95 = 0.05 $$ Now we have all the probabilities: $$\begin{array}{l|rrrrrr}x & 0 & 1 & 2 & 3 & 4 & 5 \\\\\hline p(x) & .1 & .3 & .4 & .1 & .05 & .05\end{array}$$

Construct the probability histogram

To plot the probability histogram, we will have the random variable \(x\) values on the horizontal axis and their respective probabilities \(p(x)\) on the vertical axis. Create vertical bars above each \(x\) value with heights equal to their probabilities. For example, the bar above \(x=0\) will have a height of 0.1, the bar above \(x=1\) will have a height of 0.3, and so on. The resulting histogram will visually represent the probability distribution for the random variable \(x\). Once you plot the histogram, you'll see the probability distribution for the random variable \(x\) and can use it to better understand and analyze the given data.

Key concepts

Probability Histogram

When dealing with a probability distribution, a probability histogram is an excellent visual tool that lets you see the likelihood of various outcomes of a random variable at a glance. It's a type of bar graph, where the horizontal axis represents the possible values of a random variable, and the height of each bar corresponds to the probability of that particular outcome. Constructing one is straightforward once you have all the necessary probabilities.

Based on the given exercise, for each value of the random variable x, you draw a vertical bar above the corresponding value on the horizontal axis, with its height equal to p(x), the probability of x. Thus, you will have bars of varying heights that together provide a graphical interpretation of the distribution. This visualization isn't just useful for textbooks; it's an essential tool for statisticians and data analysts as it provides immediate insights into the nature of the probability distribution, such as its symmetry, skewness, and the spread of probabilities among different outcomes.

Random Variable

A random variable, often denoted by x, Y, or another letter, is a basic concept in probability and statistics representing a quantitative variable whose value depends on the outcomes of a random phenomenon. To put it simply, it's a way of assigning numbers to each possible outcome of an experiment. There are two types of random variables: discrete and continuous. In our exercise, we are dealing with a discrete random variable, which has specific, distinct values, as shown by the integers 0 through 5.

Understanding random variables is crucial for interpreting data and making predictions. They serve as the foundation for probability calculations, and they bridge the gap between real-world random processes and mathematical modeling, allowing us to compute and work with probabilities in a structured manner.

Probability Calculations

The heart of probability theory lies in probability calculations, which allow you to quantify the likelihood of various events. The foundation of these calculations is the probability distribution of a random variable, which lists all possible values along with the chances (probabilities) of each occurring. In our scenario, we calculate the missing probability by knowing that the total must sum up to 1, since one of the outcomes must occur.

Establishing accurate probability calculations is vital as it underpins decision-making processes in diverse fields such as finance, engineering, and even everyday situations. For example, when you know that a part has a 0.05 probability of failing, you can make informed decisions about maintenance or safety checks. In a broader sense, these calculations can also be applied to complex statistics, such as predicting customer behavior, weather forecasting, and many aspects of machine learning algorithms. The real power of probability calculations lies in their ability to help us predict and plan for various scenarios, despite inherent uncertainties.