1. What Is The Expected Value Of The Probability Distribution Of The Discrete Random Variable $X$?$\[ \begin{array}{|c|c|} \hline x & P(X = X) \\ \hline 1 & 0.23 \\ \hline 3 & 0.09 \\ \hline 5 & 0.05 \\ \hline 7 & 0.01 \\ \hline 9 & 0.30

Feb 26, 2025 by ADMIN 238 views

1.1 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.

1.2 Definition of Expected Value

The expected value of a discrete random variable $X$ is denoted by $E(X)$ or $μ$ (mu). It is calculated as the sum of each possible value of $X$ multiplied by its probability of occurrence. Mathematically, it can be represented as:

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

where the sum is taken over all possible values of $X$ .

1.3 Formula for Expected Value

To calculate the expected value of a discrete random variable, we need to multiply each possible value of $X$ by its probability of occurrence and then sum up these products. The formula for expected value is:

E(X) = x_1P(X = x_1) + x_2P(X = x_2) + ... + x_nP(X = x_n)

where $x_1, x_2, ..., x_n$ are the possible values of $X$ and $P(X = x_1), P(X = x_2), ..., P(X = x_n)$ are their respective probabilities.

1.4 Example

Let's consider a discrete random variable $X$ with the following probability distribution:

$x$	$P(X = x)$
1	0.23
3	0.09
5	0.05
7	0.01
9	0.30

To calculate the expected value of $X$ , we need to multiply each possible value of $X$ by its probability of occurrence and then sum up these products.

E(X) = 1(0.23) + 3(0.09) + 5(0.05) + 7(0.01) + 9(0.30)

E(X) = 0.23 + 0.27 + 0.25 + 0.07 + 2.70

E(X) = 3.52

1.5 Interpretation of Expected Value

The expected value of a discrete random variable represents the long-run average value that the variable is expected to take on when the experiment is repeated many times. In the example above, the expected value of $X$ is 3.52, which means that if the experiment is repeated many times, the average value of $X$ is expected to be 3.52.

1.6 Properties of Expected Value

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

Linearity: The expected value of a sum of random variables is equal to the sum 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.
Non-negativity: The expected value of a non-negative random variable is non-negative.

1.7 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 $E(X) = \sum_{x} xP(X = x)$ , where the sum is taken over all possible values of $X$ . The expected value has several important properties, including linearity, homogeneity, and non-negativity.

1.8 References

[1] Ross, S. M. (2010). Introduction to Probability Models. Academic Press.
[2] Papoulis, A. (2002). Probability, Random Variables, and Stochastic Processes. McGraw-Hill.
[3] Grimmett, G. R., & Stirzaker, D. R. (2001). Probability and Random Processes. Oxford University Press.

1.9 Discussion

The expected value of a discrete random variable is an important concept in probability theory. 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 $E(X) = \sum_{x} xP(X = x)$ , where the sum is taken over all possible values of $X$ . The expected value has several important properties, including linearity, homogeneity, and non-negativity.

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 expected value and provided an example of how to calculate the expected value of a discrete random variable. The expected value is an important concept in probability theory and has many applications in statistics, engineering, and economics.

1.10 Final Thoughts

The expected value of a discrete random variable is a powerful tool for analyzing and understanding the behavior of random variables. 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 $E(X) = \sum_{x} xP(X = x)$ , where the sum is taken over all possible values of $X$ . The expected value has several important properties, including linearity, homogeneity, and non-negativity.

In conclusion, the expected value of a discrete random variable is an important concept in probability theory. 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 $E(X) = \sum_{x} xP(X = x)$ , where the sum is taken over all possible values of $X$ . The expected value has several important properties, including linearity, homogeneity, and non-negativity.

2.1 Introduction

In the 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.

2.2 Q&A

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

A: 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.

2.2.2 Q: How do I calculate the expected value of a discrete random variable?

A: To calculate the expected value of a discrete random variable, you need to multiply each possible value of the variable by its probability of occurrence and then sum up these products. The formula for expected value is:

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

where the sum is taken over all possible values of $X$ .

2.2.3 Q: What is the difference between the expected value and the mean?

A: The expected value and the mean are two related but distinct concepts. The mean is the average value of a set of data, while the expected value is the long-run average value that a random variable is expected to take on when the experiment is repeated many times.

2.2.4 Q: Can the expected value of a discrete random variable be negative?

A: Yes, the expected value of a discrete random variable can be negative. For example, if the random variable represents the amount of money lost in a game, the expected value could be negative if the game is expected to result in a loss.

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

A: The expected value of a discrete random variable represents the long-run average value that the variable is expected to take on when the experiment is repeated many times. It can be used to make predictions about the behavior of the variable and to make decisions based on those predictions.

2.2.6 Q: Can the expected value of a discrete random variable be zero?

A: Yes, the expected value of a discrete random variable can be zero. For example, if the random variable represents the number of heads obtained in a fair coin toss, the expected value would be zero since the probability of getting a head is 0.5 and the probability of getting a tail is also 0.5.

2.2.7 Q: How do I calculate the variance of a discrete random variable?

A: To calculate the variance of a discrete random variable, you need to first calculate the expected value of the variable and then calculate the squared differences between each possible value of the variable and the expected value. The formula for variance is:

Var(X) = E(X^2) - (E(X))^2

where $E(X)$ is the expected value of the variable.

2.2.8 Q: What is the relationship between the expected value and the variance of a discrete random variable?

A: The expected value and the variance of a discrete random variable are related but distinct concepts. The variance represents the spread or dispersion of the variable's distribution, while the expected value represents the central tendency of the distribution.

2.3 Conclusion

2.4 References

[1] Ross, S. M. (2010). Introduction to Probability Models. Academic Press.
[2] Papoulis, A. (2002). Probability, Random Variables, and Stochastic Processes. McGraw-Hill.
[3] Grimmett, G. R., & Stirzaker, D. R. (2001). Probability and Random Processes. Oxford University Press.

2.5 Discussion

In this article, we have answered some frequently asked questions about the expected value of a discrete random variable. We have discussed the concept of expected value, how to calculate it, and its properties. We have also provided examples and explanations to help illustrate the concepts.