The mean and variance of a Binomial(n, p) distribution are respectively 20 and 16. Then the values of n and p are

For a binomial distribution \( \text{Binomial}(n, p) \), the mean and variance are given by the following formulas:

- **Mean**: \( \mu = np \)
- **Variance**: \( \sigma^2 = np(1 - p) \)

We are given that the mean is 20 and the variance is 16. Thus, we have the following two equations:

1. \( np = 20 \)
2. \( np(1 - p) = 16 \)

Now, let's solve this system of equations step by step.

From the first equation, we have:
\[
np = 20 \quad \Rightarrow \quad p = \frac{20}{n}
\]

Substitute this value of \( p \) into the second equation:

\[
np(1 - p) = 16
\]

Substitute \( p = \frac{20}{n} \) into the equation:

\[
n \cdot \frac{20}{n} \cdot \left(1 - \frac{20}{n}\right) = 16
\]

Simplifying this expression:

\[
20 \left(1 - \frac{20}{n}\right) = 16
\]

\[
20 - \frac{400}{n} = 16
\]

\[
\frac{400}{n} = 4
\]

\[
n = 100
\]

Now that we have \( n = 100 \), substitute this value back into the first equation to find \( p \):

\[
np = 20 \quad \Rightarrow \quad 100p = 20 \quad \Rightarrow \quad p = 0.2
\]

Thus, the values of \( n \) and \( p \) are:
\[
n = 100, \quad p = 0.2
\]

Let me know if you would like any further details or clarification!

---

Here are 5 related questions for further practice:

1. What is the probability of getting exactly 25 successes in a \( \text{Binomial}(100, 0.2) \) distribution?
2. How would the mean and variance change if \( p \) were doubled in the binomial distribution?
3. How can you compute the skewness of a binomial distribution?
4. What is the probability of getting more than 30 successes in a \( \text{Binomial}(100, 0.2) \) distribution?
5. How does the binomial distribution relate to the normal distribution when \( n \) is large?

**Tip**: When \( n \) is large, the binomial distribution can often be approximated by a normal distribution using the Central Limit Theorem.

Learn how to solve a binomial distribution problem where the mean is 20 and the variance is 16. This problem involves key probability concepts, including the formulas for the mean and variance of a binomial distribution. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation