Given The Relative-frequency Table Below, Find The Expected Value Of $x$.$\[ \begin{tabular}{|c|c|} \hline $x$ & $P(X=x)$ \\ \hline 3 & 0.02 \\ \hline 5 & 0.39 \\ \hline 7 & 0.12 \\ \hline 9 & 0.47 \\ \hline \end{tabular} \\]A.
Introduction
In probability theory, the expected value of a 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. In this article, we will discuss how to find the expected value of a discrete random variable using a relative-frequency table.
What is Expected Value?
The expected value of a discrete random variable X is denoted by E(X) or μ (mu). It is calculated by multiplying each possible value of X by its probability and summing the results. Mathematically, it can be represented as:
E(X) = ∑xP(X=x)
where x represents the possible values of X and P(X=x) represents the probability of each value.
Relative-Frequency Table
A relative-frequency table is a table that shows the frequency of each value of a random variable. In this case, we are given a relative-frequency table that shows the probability of each value of X.
| x | P(X=x) |
|---|---|
| 3 | 0.02 |
| 5 | 0.39 |
| 7 | 0.12 |
| 9 | 0.47 |
Calculating the Expected Value
To calculate the expected value of X, we need to multiply each value of X by its probability and sum the results.
E(X) = (3)(0.02) + (5)(0.39) + (7)(0.12) + (9)(0.47)
Step-by-Step Calculation
Let's break down the calculation into smaller steps.
- Multiply 3 by 0.02:
(3)(0.02) = 0.06
- Multiply 5 by 0.39:
(5)(0.39) = 1.95
- Multiply 7 by 0.12:
(7)(0.12) = 0.84
- Multiply 9 by 0.47:
(9)(0.47) = 4.23
Summing the Results
Now, let's sum the results of each step.
E(X) = 0.06 + 1.95 + 0.84 + 4.23
E(X) = 7.08
Conclusion
In this article, we discussed how to find the expected value of a discrete random variable using a relative-frequency table. We calculated the expected value of X by multiplying each value of X by its probability and summing the results. The expected value of X is 7.08.
Real-World Applications
The concept of expected value has many real-world applications. For example, in finance, the expected value of a stock's return is used to determine its value. In insurance, the expected value of a claim is used to determine the premium. In engineering, the expected value of a system's performance is used to determine its reliability.
Limitations of Expected Value
While the expected value is a useful measure of central tendency, it has some limitations. For example, it does not take into account the variability of the data. It also assumes that the data is normally distributed, which may not always be the case.
Alternative Measures of Central Tendency
There are several alternative measures of central tendency that can be used in place of the expected value. These include the median and the mode. The median is the middle value of the data when it is arranged in order. The mode is the most frequently occurring value of the data.
Conclusion
In conclusion, the expected value of a discrete random variable is a measure of its central tendency. It is calculated by multiplying each value of X by its probability and summing the results. The expected value has many real-world applications, but it also has some limitations. Alternative measures of central tendency, such as the median and the mode, can be used in place of the expected value.
References
- [1] "Probability and Statistics" by James E. Gentle
- [2] "Statistics for Dummies" by Deborah J. Rumsey
- [3] "Probability and Statistics for Engineers and Scientists" by Ronald E. Walpole
Glossary
- Expected Value: The long-run average value that a random variable is expected to take on.
- Relative-Frequency Table: A table that shows the frequency of each value of a random variable.
- Probability: A measure of the likelihood of an event occurring.
- Random Variable: A variable that can take on different values with different probabilities.
- Central Tendency: A measure of the middle value of a data set.
Expected Value Q&A =====================
Frequently Asked Questions
In this article, we will answer some of the most frequently asked questions about expected value.
Q: What is expected value?
A: Expected value is a measure of the central tendency of a random variable's distribution. It represents the long-run average value that the variable is expected to take on.
Q: How is expected value calculated?
A: Expected value is calculated by multiplying each possible value of a random variable by its probability and summing the results.
Q: What is the formula for expected value?
A: The formula for expected value is:
E(X) = ∑xP(X=x)
where x represents the possible values of X and P(X=x) represents the probability of each value.
Q: What is the difference between expected value and mean?
A: Expected value and mean are often used interchangeably, but they are not exactly the same thing. The mean is a measure of the average value of a data set, while the expected value is a measure of the long-run average value of a random variable.
Q: When is expected value used?
A: Expected value is used in a variety of fields, including finance, insurance, engineering, and statistics. It is used to make predictions about the future behavior of a system or process.
Q: What are some common applications of expected value?
A: Some common applications of expected value include:
- Calculating the expected return on investment (ROI) of a stock or bond
- Determining the expected cost of a project or process
- Estimating the expected value of a claim in insurance
- Calculating the expected value of a system's performance in engineering
Q: What are some limitations of expected value?
A: Some limitations of expected value include:
- It does not take into account the variability of the data
- It assumes that the data is normally distributed, which may not always be the case
- It can be sensitive to outliers in the data
Q: What are some alternative measures of central tendency?
A: Some alternative measures of central tendency include:
- Median: The middle value of a data set when it is arranged in order
- Mode: The most frequently occurring value of a data set
- Geometric mean: The nth root of the product of n values
Q: How is expected value used in finance?
A: Expected value is used in finance to calculate the expected return on investment (ROI) of a stock or bond. It is also used to determine the expected value of a portfolio or investment.
Q: How is expected value used in insurance?
A: Expected value is used in insurance to estimate the expected value of a claim. It is also used to determine the expected cost of a policy or coverage.
Q: How is expected value used in engineering?
A: Expected value is used in engineering to calculate the expected value of a system's performance. It is also used to determine the expected cost of a project or process.
Conclusion
In conclusion, expected value is a powerful tool for making predictions about the future behavior of a system or process. It is used in a variety of fields, including finance, insurance, engineering, and statistics. While it has some limitations, it is a useful measure of central tendency that can be used to make informed decisions.
References
- [1] "Probability and Statistics" by James E. Gentle
- [2] "Statistics for Dummies" by Deborah J. Rumsey
- [3] "Probability and Statistics for Engineers and Scientists" by Ronald E. Walpole
Glossary
- Expected Value: The long-run average value that a random variable is expected to take on.
- Relative-Frequency Table: A table that shows the frequency of each value of a random variable.
- Probability: A measure of the likelihood of an event occurring.
- Random Variable: A variable that can take on different values with different probabilities.
- Central Tendency: A measure of the middle value of a data set.