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Challenge: Confidence Interval for Exponential Distribution Parameter | Estimation of Population Parameters
Probability Theory Mastering
course content

Course Content

Probability Theory Mastering

Probability Theory Mastering

1. Additional Statements From The Probability Theory
2. The Limit Theorems of Probability Theory
3. Estimation of Population Parameters
4. Testing of Statistical Hypotheses

bookChallenge: Confidence Interval for Exponential Distribution Parameter

In the previous chapter, we considered how to build confidence intervals for the parameters of a Gaussian distribution. Now let's understand how we can use this to build confidence intervals for the parameters of other distributions.
For example, let's use exponential distribution. Earlier in this course, the exponential distribution has already been used in examples. Now, let's look at this distribution in more detail and consider its PDF and characteristics:

We have to build a confidence interval for the lambda parameter to solve this problem. Let's first consider not the lambda parameter itself but the inversed parameter 1/lambda.
According to the method of moments, we can estimate this parameter using the sample mean, as we have shown earlier, such an estimate will be both non-biased and consistent.

Note

This estimate will also be effective You can verify this yourself based on the information in the previous chapter.

Confidence interval for exponential distribution parameter

Firstly, let's find the expected value and variance of our estimation:

Now let's construct the following function and analyze it:

Since we have the sum of i.i.d. random variables with a finite mathematical expectation and variance, we can apply the central limit theorem to our estimate. Then we construct the function so that its distribution does not depend on the estimated parameters. Finally, we only have to specify the analytical confidence interval.

Note

Estimates for which the Central Limit Theorem holds are called asymptotically normal. We can construct confidence intervals using the confidence intervals of the Gaussian distribution for such estimates.

Task

Thus, we analytically derived a formula by which we can calculate the confidence interval for the exponential distribution parameter, it remains to implement this in the code. You have to:

  1. Import stats module from scipy library to provide necessary calculations.
  2. Use .ppf() method to calculate z_value.
  3. Use .mean() method to get estimation of 1 / lambda parameter.

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Section 3. Chapter 9
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bookChallenge: Confidence Interval for Exponential Distribution Parameter

In the previous chapter, we considered how to build confidence intervals for the parameters of a Gaussian distribution. Now let's understand how we can use this to build confidence intervals for the parameters of other distributions.
For example, let's use exponential distribution. Earlier in this course, the exponential distribution has already been used in examples. Now, let's look at this distribution in more detail and consider its PDF and characteristics:

We have to build a confidence interval for the lambda parameter to solve this problem. Let's first consider not the lambda parameter itself but the inversed parameter 1/lambda.
According to the method of moments, we can estimate this parameter using the sample mean, as we have shown earlier, such an estimate will be both non-biased and consistent.

Note

This estimate will also be effective You can verify this yourself based on the information in the previous chapter.

Confidence interval for exponential distribution parameter

Firstly, let's find the expected value and variance of our estimation:

Now let's construct the following function and analyze it:

Since we have the sum of i.i.d. random variables with a finite mathematical expectation and variance, we can apply the central limit theorem to our estimate. Then we construct the function so that its distribution does not depend on the estimated parameters. Finally, we only have to specify the analytical confidence interval.

Note

Estimates for which the Central Limit Theorem holds are called asymptotically normal. We can construct confidence intervals using the confidence intervals of the Gaussian distribution for such estimates.

Task

Thus, we analytically derived a formula by which we can calculate the confidence interval for the exponential distribution parameter, it remains to implement this in the code. You have to:

  1. Import stats module from scipy library to provide necessary calculations.
  2. Use .ppf() method to calculate z_value.
  3. Use .mean() method to get estimation of 1 / lambda parameter.

Switch to desktopSwitch to desktop for real-world practiceContinue from where you are using one of the options below
Everything was clear?

How can we improve it?

Thanks for your feedback!

Section 3. Chapter 9
toggle bottom row

bookChallenge: Confidence Interval for Exponential Distribution Parameter

In the previous chapter, we considered how to build confidence intervals for the parameters of a Gaussian distribution. Now let's understand how we can use this to build confidence intervals for the parameters of other distributions.
For example, let's use exponential distribution. Earlier in this course, the exponential distribution has already been used in examples. Now, let's look at this distribution in more detail and consider its PDF and characteristics:

We have to build a confidence interval for the lambda parameter to solve this problem. Let's first consider not the lambda parameter itself but the inversed parameter 1/lambda.
According to the method of moments, we can estimate this parameter using the sample mean, as we have shown earlier, such an estimate will be both non-biased and consistent.

Note

This estimate will also be effective You can verify this yourself based on the information in the previous chapter.

Confidence interval for exponential distribution parameter

Firstly, let's find the expected value and variance of our estimation:

Now let's construct the following function and analyze it:

Since we have the sum of i.i.d. random variables with a finite mathematical expectation and variance, we can apply the central limit theorem to our estimate. Then we construct the function so that its distribution does not depend on the estimated parameters. Finally, we only have to specify the analytical confidence interval.

Note

Estimates for which the Central Limit Theorem holds are called asymptotically normal. We can construct confidence intervals using the confidence intervals of the Gaussian distribution for such estimates.

Task

Thus, we analytically derived a formula by which we can calculate the confidence interval for the exponential distribution parameter, it remains to implement this in the code. You have to:

  1. Import stats module from scipy library to provide necessary calculations.
  2. Use .ppf() method to calculate z_value.
  3. Use .mean() method to get estimation of 1 / lambda parameter.

Switch to desktopSwitch to desktop for real-world practiceContinue from where you are using one of the options below
Everything was clear?

How can we improve it?

Thanks for your feedback!

In the previous chapter, we considered how to build confidence intervals for the parameters of a Gaussian distribution. Now let's understand how we can use this to build confidence intervals for the parameters of other distributions.
For example, let's use exponential distribution. Earlier in this course, the exponential distribution has already been used in examples. Now, let's look at this distribution in more detail and consider its PDF and characteristics:

We have to build a confidence interval for the lambda parameter to solve this problem. Let's first consider not the lambda parameter itself but the inversed parameter 1/lambda.
According to the method of moments, we can estimate this parameter using the sample mean, as we have shown earlier, such an estimate will be both non-biased and consistent.

Note

This estimate will also be effective You can verify this yourself based on the information in the previous chapter.

Confidence interval for exponential distribution parameter

Firstly, let's find the expected value and variance of our estimation:

Now let's construct the following function and analyze it:

Since we have the sum of i.i.d. random variables with a finite mathematical expectation and variance, we can apply the central limit theorem to our estimate. Then we construct the function so that its distribution does not depend on the estimated parameters. Finally, we only have to specify the analytical confidence interval.

Note

Estimates for which the Central Limit Theorem holds are called asymptotically normal. We can construct confidence intervals using the confidence intervals of the Gaussian distribution for such estimates.

Task

Thus, we analytically derived a formula by which we can calculate the confidence interval for the exponential distribution parameter, it remains to implement this in the code. You have to:

  1. Import stats module from scipy library to provide necessary calculations.
  2. Use .ppf() method to calculate z_value.
  3. Use .mean() method to get estimation of 1 / lambda parameter.

Switch to desktopSwitch to desktop for real-world practiceContinue from where you are using one of the options below
Section 3. Chapter 9
Switch to desktopSwitch to desktop for real-world practiceContinue from where you are using one of the options below
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