Expected Value of the Distribution

Oct 13, 2025 by miodeya.com 128 views

Introduction

Annual surveys of bird nests in a certain park have been conducted to understand the distribution of the number of eggs in each nest. The random variable ${x}$ , representing the number of eggs in each nest, has been studied to determine its expected value and probability distribution. In this article, we will discuss the expected value of the distribution and the probability of a nest having at least one egg.

Expected Value of the Distribution

The expected value of a distribution is a measure of the central tendency of the data. It represents the average value that the random variable is likely to take. In this case, the expected value of the distribution was found to be 1.4 eggs. This means that, on average, each nest is expected to have 1.4 eggs.

Probability of a Nest Having at Least One Egg

The probability of a nest having at least one egg can be calculated using the expected value of the distribution. Since the expected value is 1.4 eggs, it is likely that some nests will have no eggs at all. However, the probability of a nest having at least one egg is still high.

To calculate the probability of a nest having at least one egg, we need to consider the probability distribution of the random variable ${x}$ . Let's assume that the probability distribution of ${x}$ is a Poisson distribution with a mean of 1.4 eggs. The probability of a nest having at least one egg can be calculated as follows:

P( ${x \geq 1}$ ) = 1 - P( ${x = 0}$ )

where P( ${x = 0}$ ) is the probability of a nest having no eggs.

Calculating the Probability of a Nest Having No Eggs

The probability of a nest having no eggs can be calculated using the Poisson distribution formula:

P( ${x = 0}$ ) = ${\frac{e^{-\lambda} \lambda^0}{0!}}$

where ${\lambda}$ is the mean of the distribution, which is 1.4 eggs in this case.

Substituting the value of ${\lambda}$ into the formula, we get:

P( ${x = 0}$ ) = ${\frac{e^{-1.4} (1.4)^0}{0!}}$

Simplifying the expression, we get:

P( ${x = 0}$ ) = ${e^{-1.4}}$

Calculating the Probability of a Nest Having at Least One Egg

Now that we have calculated the probability of a nest having no eggs, we can calculate the probability of a nest having at least one egg as follows:

P( ${x \geq 1}$ ) = 1 - P( ${x = 0}$ )

Substituting the value of P( ${x = 0}$ ) into the formula, we get:

P( ${x \geq 1}$ ) = 1 - ${e^{-1.4}}$

Simplifying the expression, we get:

P( ${x \geq 1}$ ) = 1 - 0.2488

P( ${x \geq 1}$ ) = 0.7512

Therefore, the probability of a nest having at least one egg is approximately 75.12%.

Conclusion

In conclusion, the expected value of the distribution of bird nests in a park was found to be 1.4 eggs. The probability of a nest having at least one egg was calculated using the Poisson distribution formula and was found to be approximately 75.12%. This means that, on average, each nest is expected to have 1.4 eggs, and the probability of a nest having at least one egg is high.

References

[1] Poisson, S. (1837). Recherches sur la probabilité des jugements en matière criminelle et en matière civile, précédées des règles générales du calcul des probabilités. Bachelier.
[2] Johnson, N. L., Kotz, S., & Kemp, A. W. (1992). Univariate discrete distributions. John Wiley & Sons.

Mathematical Formulas

Poisson distribution formula: P( ${x = k}$ ) = ${\frac{e^{-\lambda} \lambda^k}{k!}}$
Expected value formula: E( ${x}$ ) = ${\sum_{k=0}^{\infty} kP(x = k)}$
Probability of a nest having at least one egg: P( ${x \geq 1}$ ) = 1 - P( ${x = 0}$ )

Code

import math

def poisson_distribution(lambda_value, k):
    return (math.exp(-lambda_value) * (lambda_value ** k)) / math.factorial(k)

def expected_value(lambda_value):
    return sum(k * poisson_distribution(lambda_value, k) for k in range(0, 10))

def probability_at_least_one_egg(lambda_value):
    return 1 - poisson_distribution(lambda_value, 0)

lambda_value = 1.4
print("Expected value:", expected_value(lambda_value))
print("Probability of a nest having at least one egg:", probability_at_least_one_egg(lambda_value))

Q: What is the expected value of the distribution of bird nests in a park?

A: The expected value of the distribution of bird nests in a park is 1.4 eggs. This means that, on average, each nest is expected to have 1.4 eggs.

Q: What is the probability of a nest having at least one egg?

A: The probability of a nest having at least one egg is approximately 75.12%. This means that, on average, each nest is expected to have at least one egg.

Q: What is the probability of a nest having no eggs?

A: The probability of a nest having no eggs is approximately 24.88%. This means that, on average, each nest is expected to have no eggs.

Q: What is the Poisson distribution formula?

A: The Poisson distribution formula is P( ${x = k}$ ) = ${\frac{e^{-\lambda} \lambda^k}{k!}}$ , where ${\lambda}$ is the mean of the distribution and ${k}$ is the number of eggs.

Q: What is the expected value formula?

A: The expected value formula is E( ${x}$ ) = ${\sum_{k=0}^{\infty} kP(x = k)}$ , where ${P(x = k)}$ is the probability of a nest having ${k}$ eggs.

Q: How can I calculate the probability of a nest having at least one egg?

A: To calculate the probability of a nest having at least one egg, you can use the formula P( ${x \geq 1}$ ) = 1 - P( ${x = 0}$ ), where P( ${x = 0}$ ) is the probability of a nest having no eggs.

Q: What is the significance of the expected value of the distribution of bird nests in a park?

A: The expected value of the distribution of bird nests in a park is a measure of the central tendency of the data. It represents the average value that the random variable is likely to take. In this case, the expected value is 1.4 eggs, which means that, on average, each nest is expected to have 1.4 eggs.

Q: How can I use the Poisson distribution formula to calculate the probability of a nest having at least one egg?

A: To use the Poisson distribution formula to calculate the probability of a nest having at least one egg, you can substitute the value of ${\lambda}$ (the mean of the distribution) and ${k}$ (the number of eggs) into the formula P( ${x = k}$ ) = ${\frac{e^{-\lambda} \lambda^k}{k!}}$ . Then, you can calculate the probability of a nest having at least one egg using the formula P( ${x \geq 1}$ ) = 1 - P( ${x = 0}$ ).

Q: What is the relationship between the expected value and the probability of a nest having at least one egg?

A: The expected value and the probability of a nest having at least one egg are related in that the expected value represents the average value that the random variable is likely to take, while the probability of a nest having at least one egg represents the likelihood of a nest having at least one egg.

Q: How can I use the expected value formula to calculate the expected value of the distribution of bird nests in a park?

A: To use the expected value formula to calculate the expected value of the distribution of bird nests in a park, you can substitute the value of P( ${x = k}$ ) (the probability of a nest having ${k}$ eggs) into the formula E( ${x}$ ) = ${\sum_{k=0}^{\infty} kP(x = k)}$ . Then, you can calculate the expected value of the distribution of bird nests in a park.

Q: What is the significance of the probability of a nest having at least one egg?

A: The probability of a nest having at least one egg is a measure of the likelihood of a nest having at least one egg. In this case, the probability of a nest having at least one egg is approximately 75.12%, which means that, on average, each nest is expected to have at least one egg.

Q: How can I use the Poisson distribution formula to calculate the probability of a nest having no eggs?

A: To use the Poisson distribution formula to calculate the probability of a nest having no eggs, you can substitute the value of ${\lambda}$ (the mean of the distribution) and ${k = 0}$ into the formula P( ${x = k}$ ) = ${\frac{e^{-\lambda} \lambda^k}{k!}}$ . Then, you can calculate the probability of a nest having no eggs.

Q: What is the relationship between the probability of a nest having no eggs and the probability of a nest having at least one egg?

A: The probability of a nest having no eggs and the probability of a nest having at least one egg are related in that the probability of a nest having no eggs represents the likelihood of a nest having no eggs, while the probability of a nest having at least one egg represents the likelihood of a nest having at least one egg.