Let X_1,X_2,...,X_n be random sample from following probability density function

theprettyshopper7

theprettyshopper7

Answered question

2022-06-21

Answer & Explanation

alenahelenash

alenahelenash

Expert2023-05-21Added 556 answers

a) The sampling frame distribution of W=2λX¯ can be found by considering the distribution of the sample mean X¯.
Given that X1,X2,...,Xn is a random sample, the sample mean X¯ can be defined as:
X¯=1ni=1nXi
We know that each Xi follows an exponential distribution with parameter λ. The sum of exponential random variables follows a gamma distribution.
The gamma distribution with shape parameter k and scale parameter θ has the probability density function:
f(x)=1Γ(k)θkxk1exθ
In this case, since Xi follows an exponential distribution with parameter λ, the shape parameter k is n and the scale parameter θ is 1λ.
Substituting these values into the gamma distribution's probability density function, we have:
f(x)=1Γ(n)(1λ)nxn1ex1λ
Simplifying, we get:
f(x)=λnΓ(n)xn1eλx
Therefore, the sampling frame distribution of W=2λX¯ follows a gamma distribution with shape parameter n and scale parameter 2λ.
b) To construct a 100(1α)% confidence interval for λ, we can use the properties of the gamma distribution.
The gamma distribution has a relationship with the chi-squared distribution, which states that if X follows a gamma distribution with shape parameter n and scale parameter θ, then 2nX/θ2 follows a chi-squared distribution with 2n degrees of freedom.
In this case, we have W=2λX¯, which follows a gamma distribution with shape parameter n and scale parameter 2λ. Using the relationship with the chi-squared distribution, we can write:
2nX¯2λ2~χ2n2
To construct a confidence interval for λ, we can rearrange this equation:
2nX¯λ2~χ2n2
Next, we can find the critical values of the chi-squared distribution at the (1α/2) and α/2 percentiles, denoted as χα/22 and χ1α/22, respectively, with 2n degrees of freedom.
The confidence interval for λ is then given by:
(2nχ1α/221X¯,2nχα/221X¯)
This is the 100(1α)% confidence interval for λ based on the sample mean X¯.
Note that to apply this formula, we need to have the observed values of Xi to calculate the sample mean X¯ and determine the appropriate critical values from the chi-squared distribution.

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