What Is The Expected Value Of The Probability Distribution Of The Discrete Random Variable \[$ X \$\]?$\[ \begin{array}{ll} x & P(X=x) \\ \hline 2 & 0.07 \\ 4 & 0.19 \\ 6 & 0.25 \\ 8 & 0.11 \\ 10 & 0.07 \\ 12 & 0.30 \\ 14 & 0.01

Feb 26, 2025 by ADMIN 229 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. In this article, we will explore the concept of expected value and how to calculate it for a discrete random variable.

What is a Discrete Random Variable?

A discrete random variable is a variable that can take on a countable number of distinct values. In other words, the variable can only take on a specific set of values, and each value is separated by a certain amount. For example, the number of heads obtained when flipping a coin is a discrete random variable, as it can only take on the values 0, 1, or 2.

The Probability Distribution of a Discrete Random Variable

The probability distribution of a discrete random variable is a function that assigns a probability to each possible value of the variable. The probability distribution is typically represented as a table or a graph, where the x-axis represents the possible values of the variable and the y-axis represents the corresponding probabilities.

Calculating the Expected Value of a Discrete Random Variable

The expected value of a discrete random variable is calculated by multiplying each possible value of the variable by its corresponding probability and summing up the results. Mathematically, the expected value E(X) is given by:

E(X) = ∑xP(X=x)

where x represents the possible values of the variable, P(X=x) represents the corresponding probabilities, and the summation is taken over all possible values of the variable.

Example: Calculating the Expected Value of the Discrete Random Variable X

Let's consider the 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
12	0.30
14	0.01

To calculate the expected value of X, we multiply each possible value of X by its corresponding probability and sum up the results:

E(X) = (2)(0.07) + (4)(0.19) + (6)(0.25) + (8)(0.11) + (10)(0.07) + (12)(0.30) + (14)(0.01) = 0.14 + 0.76 + 1.50 + 0.88 + 0.70 + 3.60 + 0.14 = 8.32

Therefore, the expected value of the discrete random variable X is 8.32.

Properties of the Expected Value

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

Linearity: The expected value of a linear combination of discrete random variables is equal to the linear combination of their expected values.
Homogeneity: The expected value of a discrete random variable multiplied by a constant is equal to the constant times the expected value of the variable.
Non-negativity: The expected value of a discrete random variable is always non-negative.

Applications of the Expected Value

The expected value of a discrete random variable has many important applications in statistics, economics, and finance. Some examples include:

Risk management: The expected value of a discrete random variable can be used to calculate the expected loss or gain from a particular investment or business venture.
Decision-making: The expected value of a discrete random variable can be used to make decisions under uncertainty, such as choosing between different investment options or selecting the best course of action in a business situation.
Optimization: The expected value of a discrete random variable can be used to optimize systems and processes, such as determining the optimal price for a product or selecting the best production schedule.

Conclusion

In conclusion, 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 can be calculated using the formula E(X) = ∑xP(X=x), and it has several important properties, including linearity, homogeneity, and non-negativity. The expected value has many important applications in statistics, economics, and finance, and it is a fundamental concept in probability theory.

References

Ross, S. M. (2010). A First Course in Probability . 8th ed. Pearson Prentice Hall.
Wackerly, D. D., Mendenhall, W., & Scheaffer, R. L. (2008). Mathematical Statistics with Applications . 7th ed. Duxbury Press.
Kolmogorov, A. N. (1950). Foundations of the Theory of Probability . Chelsea Publishing Company.

Discussion

The expected value of a discrete random variable is a fundamental concept in probability theory, and it has many important applications in statistics, economics, and finance. The expected value can be calculated using the formula E(X) = ∑xP(X=x), and it has several important properties, including linearity, homogeneity, and non-negativity. The expected value is a measure of the central tendency of the variable's distribution, and it represents the long-run average value that the variable is expected to take on when the experiment is repeated many times.

In this article, we have discussed the concept of expected value and how to calculate it for a discrete random variable. We have also explored the properties of the expected value, including linearity, homogeneity, and non-negativity. The expected value has many important applications in statistics, economics, and finance, and it is a fundamental concept in probability theory.

Do you have any questions or comments about the expected value of a discrete random variable? Please feel free to ask or share your thoughts in the discussion section below.

Introduction

In our previous article, we discussed the concept of expected value and how to calculate it for a discrete random variable. 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 do I calculate the expected value of a discrete random variable?

A2: To calculate the expected value of a discrete random variable, you need to multiply each possible value of the variable by its corresponding probability and sum up the results. Mathematically, the expected value E(X) is given by:

E(X) = ∑xP(X=x)

where x represents the possible values of the variable, P(X=x) represents the corresponding probabilities, and the summation is taken over all possible values of the variable.

Q3: What are the properties of the expected value?

A3: The expected value of a discrete random variable has several important properties, including:

Linearity: The expected value of a linear combination of discrete random variables is equal to the linear combination of their expected values.
Homogeneity: The expected value of a discrete random variable multiplied by a constant is equal to the constant times the expected value of the variable.
Non-negativity: The expected value of a discrete random variable is always non-negative.

Q4: What are the applications of the expected value?

A4: The expected value of a discrete random variable has many important applications in statistics, economics, and finance, including:

Risk management: The expected value of a discrete random variable can be used to calculate the expected loss or gain from a particular investment or business venture.
Decision-making: The expected value of a discrete random variable can be used to make decisions under uncertainty, such as choosing between different investment options or selecting the best course of action in a business situation.
Optimization: The expected value of a discrete random variable can be used to optimize systems and processes, such as determining the optimal price for a product or selecting the best production schedule.

Q5: Can I use the expected value to predict the future value of a discrete random variable?

A5: While the expected value can provide some insight into the future value of a discrete random variable, it is not a reliable predictor of future values. The expected value is a long-run average value, and it may not reflect the actual value of the variable in a particular instance.

Q6: How do I interpret the expected value of a discrete random variable?

A6: The expected value of a discrete random variable can be interpreted in several ways, including:

Mean: The expected value of a discrete random variable is a measure of the mean or average value of the variable.
Central tendency: The expected value of a discrete random variable is a measure of the central tendency of the variable's distribution.
Long-run average: The expected value of a discrete random variable is a measure of the long-run average value that the variable is expected to take on when the experiment is repeated many times.

Q7: Can I use the expected value to compare the values of different discrete random variables?

A7: Yes, you can use the expected value to compare the values of different discrete random variables. The expected value can provide a common metric for comparing the values of different variables, and it can help you to make informed decisions under uncertainty.

Q8: What are some common mistakes to avoid when calculating the expected value of a discrete random variable?

A8: Some common mistakes to avoid when calculating the expected value of a discrete random variable include:

Forgetting to multiply each possible value of the variable by its corresponding probability
Forgetting to sum up the results
Using an incorrect formula or method for calculating the expected value

Conclusion

In conclusion, the expected value of a discrete random variable is a fundamental concept in probability theory, and it has many important applications in statistics, economics, and finance. By understanding the concept of expected value and how to calculate it, you can make informed decisions under uncertainty and optimize systems and processes.

References

Ross, S. M. (2010). A First Course in Probability . 8th ed. Pearson Prentice Hall.
Wackerly, D. D., Mendenhall, W., & Scheaffer, R. L. (2008). Mathematical Statistics with Applications . 7th ed. Duxbury Press.
Kolmogorov, A. N. (1950). Foundations of the Theory of Probability . Chelsea Publishing Company.

Discussion

Do you have any questions or comments about the expected value of a discrete random variable? Please feel free to ask or share your thoughts in the discussion section below.