Le Y is a random variable that follows a binomial distribution with parameters n, pas X~ Binom(n, p). Let ê Y + 1 be an estimator of the parameter p. N Find the bias of the estimator .
Le Y is a random variable that follows a binomial distribution with parameters n, pas X~ Binom(n, p). Let ê Y + 1 be an estimator of the parameter p. N Find the bias of the estimator .
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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![### Understanding Estimators in Binomial Distributions: A Problem Solving Approach
Let's explore an interesting problem related to estimating parameters in binomial distributions.
Given the following:
\( Y \) is a random variable that follows a binomial distribution with parameters \( n \) and \( p \). This can be mathematically denoted as \( X \sim \text{Binom}(n, p) \).
We define the estimator \(\hat{\theta}\) for the parameter \( p \) as:
\[
\hat{\theta} = \frac{Y}{n} + 1
\]
The task is to find the bias of the estimator \(\hat{\theta}\).
### Steps to Find the Bias of the Estimator
1. **Understand the Definition of Bias**: The bias of an estimator is the difference between the expected value of the estimator and the true value of the parameter being estimated.
Mathematically, for an estimator \(\hat{\theta}\) of a parameter \( \theta \), the bias is given by:
\[
\text{Bias}(\hat{\theta}) = E(\hat{\theta}) - \theta
\]
2. **Calculate the Expected Value**: First, we need to determine the expected value of the estimator \(\hat{\theta}\).
3. **Use the Properties of Expectation**: Since \( Y \) is a binomial random variable with parameters \( n \), \( p \):
\[
E\left( \frac{Y}{n} \right) = \frac{E(Y)}{n} = \frac{np}{n} = p
\]
Then substituting back into the estimator:
\[
E(\hat{\theta}) = E\left( \frac{Y}{n} + 1 \right) = E\left( \frac{Y}{n} \right) + E(1) = p + 1
\]
4. **Determine the Bias**:
\[
\text{Bias}(\hat{\theta}) = E(\hat{\theta}) - p = (p + 1) - p = 1
\]
Thus, the bias of the estimator \(\hat{\theta}\) is 1.
This exercise exemplifies how to compute the bias of an estimator and is a crucial skill in statistical analysis, allowing us to evaluate the accuracy and reliability of](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffb989860-4d1c-4fb5-9e6b-42a4528dce9c%2Fb3366af7-2c27-4cd5-8c0b-e1efd138821e%2F0vqdo9p_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Understanding Estimators in Binomial Distributions: A Problem Solving Approach
Let's explore an interesting problem related to estimating parameters in binomial distributions.
Given the following:
\( Y \) is a random variable that follows a binomial distribution with parameters \( n \) and \( p \). This can be mathematically denoted as \( X \sim \text{Binom}(n, p) \).
We define the estimator \(\hat{\theta}\) for the parameter \( p \) as:
\[
\hat{\theta} = \frac{Y}{n} + 1
\]
The task is to find the bias of the estimator \(\hat{\theta}\).
### Steps to Find the Bias of the Estimator
1. **Understand the Definition of Bias**: The bias of an estimator is the difference between the expected value of the estimator and the true value of the parameter being estimated.
Mathematically, for an estimator \(\hat{\theta}\) of a parameter \( \theta \), the bias is given by:
\[
\text{Bias}(\hat{\theta}) = E(\hat{\theta}) - \theta
\]
2. **Calculate the Expected Value**: First, we need to determine the expected value of the estimator \(\hat{\theta}\).
3. **Use the Properties of Expectation**: Since \( Y \) is a binomial random variable with parameters \( n \), \( p \):
\[
E\left( \frac{Y}{n} \right) = \frac{E(Y)}{n} = \frac{np}{n} = p
\]
Then substituting back into the estimator:
\[
E(\hat{\theta}) = E\left( \frac{Y}{n} + 1 \right) = E\left( \frac{Y}{n} \right) + E(1) = p + 1
\]
4. **Determine the Bias**:
\[
\text{Bias}(\hat{\theta}) = E(\hat{\theta}) - p = (p + 1) - p = 1
\]
Thus, the bias of the estimator \(\hat{\theta}\) is 1.
This exercise exemplifies how to compute the bias of an estimator and is a crucial skill in statistical analysis, allowing us to evaluate the accuracy and reliability of
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