Probability Distribution Calculation For Random Variable U

Introduction to Probability Distributions

In the realm of probability and statistics, understanding the behavior of random variables is crucial. A random variable is essentially a variable whose value is a numerical outcome of a random phenomenon. These variables can be discrete, taking on a finite or countably infinite number of values, or continuous, assuming any value within a given range. The probability distribution of a random variable provides a complete description of its possible values and the likelihood of each value occurring. This article delves into the concept of probability distributions, focusing on a discrete random variable U and its associated probabilities. We will explore how to determine missing probabilities and calculate probabilities of specific events involving the random variable.

The probability distribution is a cornerstone concept in understanding the behavior of random variables. It serves as a comprehensive map, delineating all possible values that a random variable can assume, coupled with the likelihood or probability of each value's occurrence. This distribution is not merely a theoretical construct; it is a fundamental tool for prediction, analysis, and decision-making across various fields, from finance and engineering to social sciences and healthcare. For discrete random variables, the probability distribution is often represented as a table or a function that assigns probabilities to each specific value the variable can take. The sum of all probabilities in a discrete distribution must equal 1, reflecting the certainty that one of the possible outcomes will occur. Understanding the shape and characteristics of a probability distribution allows us to infer the central tendencies, variability, and potential extreme values of the random variable, providing valuable insights for statistical inference and real-world applications. In this exploration, we aim to demystify the concept of probability distributions, emphasizing their practical significance and analytical power in deciphering random phenomena.

Problem Statement

Consider a discrete random variable U that can take on the values 0, 1, 2, 3, and 4. The probabilities associated with the values 0, 1, 2, and 3 are given in the table below. However, the probability associated with the value 4 is missing, presenting our first challenge. Our objective is twofold: first, to determine the missing probability Pr(U = 4), and second, to calculate the probability that U is greater than or equal to 2, denoted as Pr(U ≥ 2). These tasks will allow us to fully characterize the probability distribution of U and understand its behavior.

(a) Determine the probability that U = 4.

(b) Find Pr(U ≥ 2).

(a) Determining the Probability Pr(U = 4)

To determine the probability that U equals 4, Pr(U = 4), we rely on a fundamental principle of probability distributions: the sum of the probabilities of all possible outcomes must equal 1. This principle stems from the fact that when we conduct a random experiment, one of the possible outcomes must occur. Therefore, the collective probability of all these outcomes represents the certainty of an event happening, which is numerically expressed as 1. In the context of our discrete random variable U, this means that the probabilities associated with each value it can take (0, 1, 2, 3, and 4) must add up to 1.

Mathematically, this can be expressed as:

Pr(U = 0) + Pr(U = 1) + Pr(U = 2) + Pr(U = 3) + Pr(U = 4) = 1

We are given the probabilities for U = 0, U = 1, U = 2, and U = 3. By substituting these values into the equation, we can solve for the unknown probability Pr(U = 4). This approach leverages the axiomatic definition of probability, where the total probability space is normalized to 1, ensuring a consistent and coherent framework for probability calculations. The principle of summing probabilities to 1 is not only a mathematical necessity but also an intuitive reflection of how we perceive certainty in random phenomena. When we know the probabilities of all other outcomes, this principle provides a powerful tool for deducing the probability of the remaining outcome, as demonstrated in this case with Pr(U = 4).

Substituting the given probabilities, we have:

(1/17) + (2/17) + (4/17) + (8/17) + Pr(U = 4) = 1

Combining the fractions, we get:

(1 + 2 + 4 + 8)/17 + Pr(U = 4) = 1

15/17 + Pr(U = 4) = 1

Now, to isolate Pr(U = 4), we subtract 15/17 from both sides of the equation:

Pr(U = 4) = 1 - 15/17

Pr(U = 4) = (17/17) - (15/17)

Pr(U = 4) = 2/17

Therefore, the probability that U equals 4 is 2/17. This result completes the probability distribution of U, providing us with a comprehensive understanding of the likelihood of each possible value. With this information, we can now proceed to calculate other probabilities, such as the probability that U is greater than or equal to 2.

(b) Finding Pr(U ≥ 2)

Now, let's turn our attention to the second part of the problem: finding the probability that U is greater than or equal to 2, denoted as Pr(U ≥ 2). This probability represents the likelihood that U will take on a value of 2, 3, or 4. To calculate this, we can leverage the principle of additivity of probabilities for mutually exclusive events. In simpler terms, if we have several distinct outcomes, the probability of any one of them occurring is the sum of their individual probabilities. In the case of our random variable U, the events U = 2, U = 3, and U = 4 are mutually exclusive, meaning that U cannot simultaneously take on two different values.

The mathematical formulation for Pr(U ≥ 2) is as follows:

Pr(U ≥ 2) = Pr(U = 2) + Pr(U = 3) + Pr(U = 4)

We already know the probabilities for U = 2, U = 3, and we have just calculated the probability for U = 4 in the previous section. By summing these probabilities, we can directly determine Pr(U ≥ 2). This approach not only simplifies the calculation but also highlights the importance of understanding the properties of probability distributions, particularly the additivity principle. The ability to decompose complex probability calculations into sums of simpler probabilities is a powerful tool in statistical analysis and decision-making. Furthermore, understanding Pr(U ≥ 2) provides valuable insights into the behavior of the random variable U, indicating the likelihood of U falling within a specific range of values. This information can be crucial in various applications, such as risk assessment, forecasting, and statistical inference.

We have the following probabilities:

• Pr(U = 2) = 4/17
• Pr(U = 3) = 8/17
• Pr(U = 4) = 2/17

Substituting these values into the equation, we get:

Pr(U ≥ 2) = (4/17) + (8/17) + (2/17)

Pr(U ≥ 2) = (4 + 8 + 2)/17

Pr(U ≥ 2) = 14/17

Therefore, the probability that U is greater than or equal to 2 is 14/17. This result, along with the probability Pr(U = 4) calculated earlier, provides a complete picture of the probability distribution of the random variable U, allowing for further analysis and interpretation.

Conclusion

In conclusion, we have successfully determined the missing probability Pr(U = 4) and calculated the probability Pr(U ≥ 2) for the given probability distribution of the discrete random variable U. By applying the fundamental principle that the sum of all probabilities must equal 1, we found that Pr(U = 4) is 2/17. Furthermore, by utilizing the additivity of probabilities for mutually exclusive events, we calculated Pr(U ≥ 2) to be 14/17. These results not only answer the specific questions posed but also demonstrate the importance of understanding the properties of probability distributions. The ability to manipulate probabilities, calculate missing values, and determine probabilities of events is crucial in various fields, from statistics and mathematics to real-world applications in finance, engineering, and decision-making. This exercise provides a solid foundation for further exploration of probability theory and its applications.

Understanding probability distributions is essential for making informed decisions in the face of uncertainty. By mastering the concepts and techniques presented in this article, readers can enhance their ability to analyze and interpret data, predict outcomes, and make sound judgments in a variety of contexts. The random variable U served as a practical example to illustrate key principles, but the methodologies discussed are applicable to a wide range of scenarios involving discrete random variables. As we continue to navigate an increasingly complex world, a strong grasp of probability and statistics will undoubtedly be a valuable asset.