Definition of Expected Value

Oct 13, 2025 by thesweetdesigns.com 111 views

Introduction

In probability theory, the expected value of a discrete random variable is a measure of the central tendency of the variable's distribution. It represents the long-run average value that the variable is expected to take on when the experiment is repeated many times. The expected value is a fundamental concept in statistics and is used in a wide range of applications, including finance, engineering, and economics.

Definition of Expected Value

The expected value of a discrete random variable ${$ X$}$ is denoted by ${$ E(X)$}$ and is defined as the sum of the product of each possible value of ${$ X$}$ and its corresponding probability. Mathematically, it can be represented as:

E(X) = \sum_{x} xP(X = x)

where ${$ x$}$ represents each possible value of ${$ X$}$ and ${$ P(X = x)$}$ represents the probability of each value.

Formula for Expected Value

To calculate the expected value of a discrete random variable, we need to follow these steps:

List all possible values of the variable and their corresponding probabilities.
Multiply each value by its probability.
Sum up the products obtained in step 2.

For example, let's consider a discrete random variable ${$ X$}$ with the following probability distribution:

${$ X$}$	${$ P(X = x)$}$
2	0.07
4	0.19
6	0.25
8	0.11
10	0.07

To calculate the expected value of ${$ X$}$, we multiply each value by its probability and sum up the products:

E(X) = (2)(0.07) + (4)(0.19) + (6)(0.25) + (8)(0.11) + (10)(0.07)

E(X) = 0.14 + 0.76 + 1.5 + 0.88 + 0.7

E(X) = 3.98

Properties of Expected Value

The expected value of a discrete random variable has several important properties, including:

Linearity: The expected value of a linear combination of random variables is equal to the linear combination of their expected values.
Homogeneity: The expected value of a random variable multiplied by a constant is equal to the constant times the expected value of the random variable.
Additivity: The expected value of the sum of two or more random variables is equal to the sum of their expected values.

Applications of Expected Value

The expected value of a discrete random variable has numerous applications in various fields, including:

Finance: Expected value is used to calculate the expected return on investment, which helps investors make informed decisions.
Engineering: Expected value is used to design and optimize systems, such as communication networks and supply chains.
Economics: Expected value is used to model economic systems and make predictions about future outcomes.

Conclusion

In conclusion, the expected value of a discrete random variable is a fundamental concept in probability theory that represents the long-run average value of the variable's distribution. It is calculated by summing up the product of each possible value and its corresponding probability. The expected value has several important properties, including linearity, homogeneity, and additivity, and has numerous applications in finance, engineering, and economics.

References

[1] Ross, S. M. (2010). Introduction to Probability Models. Academic Press.
[2] Johnson, N. L., Kotz, S., & Balakrishnan, N. (1995). Continuous Univariate Distributions. John Wiley & Sons.
[3] Papoulis, A. (1984). Probability, Random Variables, and Stochastic Processes. McGraw-Hill.

Discussion

The expected value of a discrete random variable is a powerful tool for analyzing and modeling complex systems. It provides a way to quantify the central tendency of a variable's distribution and make predictions about future outcomes. However, it is essential to note that the expected value is not always the best measure of central tendency, and other measures, such as the median and mode, may be more appropriate in certain situations.

Example Problems

A discrete random variable ${$ X$}$ has the following probability distribution:

${$ X$}$	${$ P(X = x)$}$
1	0.2
2	0.3
3	0.5

Calculate the expected value of ${$ X$}$.

A discrete random variable ${$ X$}$ has the following probability distribution:

${$ X$}$	${$ P(X = x)$}$
5	0.1
10	0.2
15	0.7

Calculate the expected value of ${$ X$}$.

Solutions

To calculate the expected value of ${$ X$}$, we multiply each value by its probability and sum up the products:

E(X) = (1)(0.2) + (2)(0.3) + (3)(0.5)

E(X) = 0.2 + 0.6 + 1.5

E(X) = 2.3

To calculate the expected value of ${$ X$}$, we multiply each value by its probability and sum up the products:

E(X) = (5)(0.1) + (10)(0.2) + (15)(0.7)

E(X) = 0.5 + 2 + 10.5

E(X) = 13$$<br/> # Q&A: Expected Value of a Discrete Random Variable

Introduction

In our previous article, we discussed the expected value of a discrete random variable and its importance in probability theory. In this article, we will answer some frequently asked questions about the expected value of a discrete random variable.

Q1: What is the expected value of a discrete random variable?

A1: The expected value of a discrete random variable is a measure of the central tendency of the variable's distribution. It represents the long-run average value that the variable is expected to take on when the experiment is repeated many times.

Q2: How is the expected value of a discrete random variable calculated?

A2: The expected value of a discrete random variable is calculated by summing up the product of each possible value and its corresponding probability. Mathematically, it can be represented as: