Question

Use the probability distribution for the random variable \(x\) to answer the questions. $$\begin{array}{l|lllll}x & 0 & 1 & 2 & 3 & 4 \\\\\hline p(x) & .1 & .3 & .3 & ? & .1\end{array}$$ What is the probability that \(x\) is 3 or less?

Short answer

Answer: The probability that x is 3 or less is 0.9.

Step-by-step solution

Find the probability of x=3

To find the probability of \(x=3\), we need to consider the sum of the probabilities of all possible outcomes of the random variable \(x\). Since the sum must equal 1, we can write an equation and solve for the probability of \(x=3\): $$0.1+0.3+0.3+p(3)+0.1=1$$ Rearranging the equation, we get: $$p(3)=1-0.1-0.3-0.3-0.1$$ Solving for \(p(3)\), we get: $$p(3)=0.2$$ Now that we have found the probability of \(x=3\), we can move on to compute the required probability.

Compute the probability of x is 3 or less

To find the probability that \(x\) is 3 or less, we simply sum the probabilities of \(x=0\), \(x=1\), \(x=2\), and \(x=3\). From the given table, we have: $$p(x\le3) = p(0)+p(1)+p(2)+p(3)$$ By plugging in the corresponding values from the table, we get: $$p(x\le3) = 0.1+0.3+0.3+0.2$$ Finally, we compute the answer: $$p(x\le3) = 0.9$$ So, the probability that \(x\) is 3 or less is 0.9.

Key concepts

Random Variable

In the realm of statistics and probability, a random variable is a fundamental concept that serves as the cornerstone for various calculations. It's not a regular variable you might encounter in algebra; instead, it represents a quantity whose value is determined by the outcome of a random phenomenon. In other words, it's a numerical description of the outcomes of some random process. For example, when you roll a fair six-sided die, the outcome (ranging from 1 to 6) is a random variable.

Random variables can be discrete or continuous. A discrete random variable takes on a countable number of distinct values, like the result of a die roll or the exercise's variable, which can be 0, 1, 2, 3, or 4. Each of these outcomes has an associated probability, denoted as p(x), which represents the likelihood of its occurrence. The sum of all the probabilities for a discrete random variable must equal 1, since one of the outcomes must occur.

When dealing with random variables, it is essential to establish the different outcomes as well as their corresponding probabilities to form what is referred to as the probability distribution. This distribution provides a comprehensive view of all possible outcomes and their probabilities, thus laying the groundwork for further statistical analysis.

Probability Calculation

The probability calculation is a vital process that quantifies the likelihood of an event occurring. In essence, it's all about numbers and how these numbers can articulate the chance of different outcomes in a random variable scenario. Probabilities are expressed as numbers between 0 and 1, with 0 indicating that an event is impossible, and 1 indicating that an event is certain to occur.

When the probability distribution is provided, as in the exercise, the computation becomes a task of addition. To calculate the probability of a particular event, such as the random variable being equal to or less than 3, you add up the probabilities for all outcomes that fall within the event's description. In the context of our exercise, this meant adding the probabilities of the variable being 0, 1, 2, or 3. This type of aggregation follows the Additive Rule of Probability, which states that the probability of one event or another occurring is the sum of their respective probabilities, provided the events are mutually exclusive.

A common mistake some students make is overlooking the requirement for the probabilities to sum up to 1. Ensuring this constraint provides a check on the accuracy of the given probabilities and prevents computational errors in probability calculation.

Cumulative Probability

When you hear about cumulative probability, think of it as 'adding up' probabilities incrementally to receive a 'running total' probability. It's the probability that a random variable is less than or equal to a certain value. In other words, it's the sum of the probabilities for all outcomes up to and including that value.

To illustrate, consider the exercise given where the cumulative probability for the variable being 3 or less is calculated. Here's how you can interpret the calculation process:

• You start with the smallest possible value of the random variable.
• Next, add the probability of the next higher value.
• Continue this process until you've included the probability of the value in question, in this case, 3.

Doing so gives us a sense of 'how much' of the distribution we've accounted for up to a certain point.

This concept is broadly utilized in statistical inference, forecasting, and risk assessment. Understanding cumulative probability is critical, not just for textbook exercises but also for real-world applications where one might need to assess the probability of an event occurring up to a certain point. For instance, in quality control, determining the probability that a manufactured product's dimensions are within a specified tolerance range.