Finding Gamma Distribution Confidence Interval

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In summary, to find a (1-alpha)% confidence interval for the unknown scale parameter beta in a gamma distribution, we can use the Minimum Variance Unbiased Estimator (MVUE) beta hat = Ybar/alpha and the chi-squared distribution. The critical value zα/2 can be found by solving P(Z > zα/2) = α/2, where Z is a chi-squared distributed random variable with n-1 degrees of freedom. The confidence interval for beta is then [Ybar/alpha ± zα/2*Sqrt(Ybar/(alpha*n))].
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renolovexoxo
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I've been working on this problem for a while, but I am really not sure that I'm doing it right. Here is the statement:

Let Y1; Y2; : : : ; Yn be i.i.d. from a gamma distribution with known shape parameter alpha and unknownscale parameter beta. Find a (1-alpha )% condfidence interval for the parameter . (Hint: the Minimum Variance Unbiased Estimator for beta is beta hat = Ybar/alpha )

I've attached the work that I've done, but I haven't used the MVUE, so I feel like I made a mistake in my distribution somewhere. Can anyone help me out?
hw.jpg
 
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The answer to this problem is to use the MVUE and the chi-squared distribution. The MVUE for beta is β^ = Ybar/α, where Ybar is the sample mean. To construct the (1-α)% confidence interval, we need to find the critical value zα/2 such that P(Z > zα/2) = α/2, where Z is a chi-squared distributed random variable with n – 1 degrees of freedom. Then the confidence interval for beta is given by: [Ybar/α ± zα/2*Sqrt(Ybar/(α*n))] where Sqrt(Ybar/(α*n)) is the standard error of the estimate.
 

1. What is a gamma distribution?

A gamma distribution is a type of probability distribution that is commonly used to model continuous data with positive and skewed values. It is characterized by two parameters, shape and scale, and is often used to model waiting times or the time until a certain event occurs.

2. What is a confidence interval?

A confidence interval is a range of values that is used to estimate the true value of a population parameter with a certain level of confidence. It is typically calculated from a sample of data and is used to make inferences about the population.

3. How do you find a confidence interval for a gamma distribution?

To find a confidence interval for a gamma distribution, you will need to know the sample mean, sample size, and the shape and scale parameters of the distribution. You can then use a statistical software or a calculator to calculate the confidence interval using a formula specific to the gamma distribution.

4. What is the significance of a confidence interval for a gamma distribution?

A confidence interval for a gamma distribution can help you determine the range of values in which the true population parameter is likely to fall with a certain level of confidence. This can be useful in making decisions or drawing conclusions about the population based on a sample of data.

5. How can a gamma distribution confidence interval be interpreted?

A gamma distribution confidence interval can be interpreted as a range of values that is likely to contain the true mean of a population with a certain level of confidence. For example, a 95% confidence interval for a gamma distribution with a mean of 10 might be 8 to 12, meaning that we are 95% confident that the true mean of the population falls within this range.

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