Expected Value (Weighted Average) in FE Electrical Exam

In electrical engineering, precision and accuracy are paramount. From designing complex circuits to optimizing power systems, every decision must carefully consider potential outcomes and their probabilities.

One key concept that plays a pivotal role in this process is the Expected Value, also known as the Weighted Average. In the realm of the Fundamentals of Engineering (FE) Electrical Exam, mastering this concept is not only crucial for success but also holds the potential to unlock new horizons in the field.

At its core, Expected Value is a mathematical tool that enables engineers to make informed decisions by quantifying the anticipated outcomes of various scenarios. A weighted average is obtained by assigning probabilities to each possible outcome and multiplying them by their respective values, offering a glimpse into the most probable result.

This seemingly magical ability to peek into the future allows electrical engineers to assess risks, optimize designs, and evaluate potential solutions with unprecedented confidence. This blog post will discuss why Expected Value in the FE Electrical Exam is a keep topic and how it helps solve complex problems efficiently and effectively.

Let’s dive deep into the details.

Importance of Expected Value in Electrical Engineering

• Minimizing Risk, Maximizing Reliability – Expected Value empowers electrical engineers to assess and minimize risks associated with design decisions. By quantifying the potential outcomes and their probabilities, engineers can make informed choices that maximize reliability and reduce the likelihood of failures or errors in electrical systems.
• Optimizing Resource Allocation – Electrical engineering projects often involve allocating limited resources such as time, budget, and materials. Expected Value helps engineers optimize these allocations by evaluating the potential outcomes and their associated values. This enables them to make resource allocation decisions that yield the highest expected return or benefit.
• Enhancing Decision-Making – Expected Value provides a systematic framework for decision-making in electrical engineering. It enables engineers to evaluate their expected outcomes by comparing different options or solutions. By considering the probabilities and values associated with each option, engineers can select the most favorable course of action, leading to efficient and effective problem-solving.
• Evaluating Cost-Effectiveness – Cost is a critical consideration in electrical engineering projects. Expected Value helps engineers assess the cost-effectiveness of various design choices or solutions. By weighing the desired outcomes against the associated costs, engineers can determine the most cost-effective approach that balances performance and budgetary constraints.
• Enabling Risk-Benefit Analysis – Electrical engineering projects often involve trade-offs between risks and benefits. Expected Value facilitates a comprehensive risk-benefit analysis by quantifying the potential outcomes and their probabilities. Engineers can then weigh the anticipated benefits against the potential risks, enabling them to make informed decisions that balance innovation and safety.

Calculation of Expected Value

The expected value, also known as the mean or average, is a concept in probability theory that represents the long-term average of a random variable. It provides a way to quantify the central tendency of a distribution of values.

In the context of a weighted average, each value is assigned a weight representing its relative importance or probability of occurrence. The expected value of a weighted average is calculated by multiplying each value by its corresponding weight and summing up the results.

Mathematically, the expected value of a weighted average is given by:

E(X) = wx + wx + … + wₙxₙ

Where:

E(X) is the expected value of the weighted average.

x, x, …… are the individual values.

w, w, …… are the corresponding weights assigned to each value.

To calculate the expected value, multiply each value by its weight and sum up the results. Based on the given weights, the result estimates the “typical” value you would expect to obtain.

Discrete Random Variable

A discrete random variable is a type of random variable that represents values that can be counted or enumerated. It has a countable or finite number of possible values associated with a certain probability. Discrete random variables are typically used to model situations where outcomes are distinct and separate.

In the plant, you have a set of switches that control various devices. Each switch can be either “on” or “off.” You investigate the number of devices turned “on” at any given time as a discrete random variable.

You observe the state of each switch periodically and record the number of devices turned “on.” Based on your data, you create a probability distribution that assigns probabilities to different counts of devices “on.” For example, the probability of having 3 devices “on” might be higher than having 5 devices “on.”

Analyzing the discrete random variable gives you insights into the power consumption patterns associated with different device states. This information allows you to optimize energy usage, identify energy-intensive periods, and make informed decisions about power management in the plant.

Continuous Random Variable

A continuous random variable is a type of random variable that represents values that fall within a range. It has an uncountable or infinite number of possible values, and these values can take any real number within a specified interval. Continuous random variables model situations where outcomes vary continuously and smoothly.

Next, you want to examine the power consumption of a specific device in the plant. You install sensors to measure the device’s power usage continuously over time.

As you collect data, you observe that the power consumption values are not limited to specific discrete levels. Instead, they form a continuous range of values. For instance, the power consumption might fluctuate smoothly between 100 kW and 200 kW.

To analyze this continuous random variable, you construct a probability density function (PDF) that describes the likelihood of different power consumption values occurring. The PDF provides a smooth curve that characterizes the device’s power consumption behavior.

By understanding the continuous random variable representing power consumption, you can estimate the average power usage, identify peak demand periods, and design efficient energy management strategies.

Let’s dive into the detailed calculation of expected value for discrete and continuous random variables and discuss some properties of expected value. We’ll use an example from electrical engineering to illustrate the concepts.

Expected Value for Discrete Random Variables

Suppose we have a discrete random variable X that represents the voltage output of an electrical circuit. The possible values of X are 2V, 4V, and 6V, and their corresponding probabilities are 0.3, 0.4, and 0.3, respectively.

To calculate the expected value (E(X)) for this discrete random variable, we use the formula:

E(X) = xp + xp + xp

Where:

x, x, x are the possible values of X (2V, 4V, 6V in this case).

p, p, p are the corresponding probabilities of each value.

Substituting the values from our example:

E(X) = (2V)(0.3) + (4V)(0.4) + (6V)(0.3)

= 0.6V + 1.6V + 1.8V

= 4V

Therefore, the expected value of the voltage output is 4V, which represents the long-term average voltage we would expect to obtain.

Expected Value for Continuous Random Variables

Now, let’s consider a continuous random variable Y representing the current flowing through an electrical component. The probability density function (PDF) of Y is given by:

f(y) = 2y for 0 ≤ y ≤ 1

To calculate the expected value (E(Y)) for this continuous random variable, we use the integral:

E(Y) = ∫[0 to 1] y * f(y) dy

Substituting the PDF into the integral:

E(Y) = ∫[0 to 1] y * 2y dy

= 2 ∫[0 to 1] y² dy

= 2 * [y³/3] [0 to 1]

= 2 * (1/3 – 0)

= 2/3

Therefore, the expected value of the current is 2/3, representing the long-term average current we would expect to obtain.

Differences between Discrete and Continuous Random Variables

Countable vs. Uncountable Values

Discrete random variables have countable or finite values, while continuous random variables have uncountable or infinite values.

In the example, the discrete random variable represents the count of devices “on,” which has a limited number of possible values (0, 1, 2, …). The continuous random variable represents the power consumption, which can take any value within a range.

Distinct Outcomes vs. Continuous Variation

Discrete random variables model situations where outcomes are distinct and separate, while continuous random variables represent situations with a continuous and smooth variation.

In the example, the discrete random variable differentiates between the states of the switches (devices “on” or “off”). The continuous random variable captures the continuous fluctuations in power consumption values.

Understanding the differences between discrete and continuous random variables allows engineers to select appropriate mathematical models and techniques to analyze and solve problems in electrical engineering effectively.

Here’s a tabular comparison of discrete and continuous random variables:

Suppose we have a random variable X representing the number of faulty components in an electrical system. The probability distribution of X is as follows:

In this problem, we deal with a discrete random variable representing the count of faulty components. The probabilities are assigned to specific values (0, 1, and 2). The problem cannot be solved using the continuous random variable method because continuous random variables deal with values that fall within a range rather than specific counts.

To calculate the expected value (E(X)) for a discrete random variable, we use the formula:

E(X) = xp + xp+ xp

Substituting the values from the table:

E(X) = (0)(0.2) + (1)(0.5) + (2)(0.3)

= 0 + 0.5 + 0.6

= 1.1

Therefore, the expected value of the number of faulty components is 1.1.

*It’s important to note that for continuous random variables, we use probability density functions (PDFs) and integrals to calculate the expected value, which is not applicable in this case.

*By differentiating between discrete and continuous random variables, engineers can appropriately choose the appropriate methods for analyzing and solving problems based on the nature of the variables involved in electrical engineering scenarios.

Applications of Expected Value in Electrical Engineering

The expected value in the FE electrical exam can help you prepare for a promising career as an electrical engineer. It has a lot of potential applications beyond your academics.

Some of these applications and industry-specific use cases of techniques you studied for Expected value in the FE electrical exam preparation include:

1. Reliability Analysis

Requirement

Reliability analysis in electrical engineering involves assessing the likelihood of system failures and optimizing designs for enhanced reliability.

Use-case

Expected Value plays a crucial role in reliability analysis by quantifying the probabilities of component failures and their impact on system performance. Engineers can assign probabilities to various failure modes, calculate the expected value of system failure, and use this information to make design decisions.

For example, engineers can determine the expected value of system downtime in power distribution systems and use it to select components with higher reliability ratings, ensuring an uninterrupted power supply.

2. Signal Processing

Requirement

Signal processing aims to extract valuable information from electrical signals and improve their quality, accuracy, and efficiency.

Use-case

Expected Value is utilized in signal processing to optimize algorithms and filter design. Engineers can assign probabilities to signal characteristics, such as noise levels or distortion, and calculate the expected signal quality value.

This information helps select appropriate filters or processing techniques to enhance the desired signal characteristics while minimizing unwanted noise. For instance, engineers can use expected value calculations in audio signal processing to design filters that reduce background noise without distorting the desired audio signal.

3. Control Systems

Requirement

Control systems regulate and manage electrical systems to achieve desired performance and stability.

Use-case

Expected Value plays a crucial role in control systems by aiding in system modeling, parameter estimation, and optimization. Engineers can assign probabilities to uncertainties in system parameters, such as sensor measurements or actuator responses, and calculate the expected value of system behavior.

This information helps design robust controllers that can effectively handle uncertainties and disturbances. For example, in autonomous vehicle control, expected value calculations can estimate the probability of different driving scenarios and guide decision-making algorithms to ensure safe and efficient operation.

4. Queuing Theory

Requirement

Queuing theory analyzes and optimizes waiting for lines or queues in electrical systems, such as communication networks or manufacturing processes.

Use-case

Expected Value is extensively used in queuing theory to evaluate system performance measures, such as average waiting time or queue length. Engineers can assign probabilities to different arrival rates and service times and calculate the expected value of system performance metrics.

This information aids in optimizing system parameters, such as server capacity or buffer size, to minimize waiting times and improve overall efficiency. For example, expected value calculations in telecommunications networks help determine the optimal number of channels or bandwidth allocation to ensure efficient data transmission and minimize queuing delays.