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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¯=1n∑i=1nXiWe 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)θkxk−1e−xθIn 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&#039;s probability density function, we have:f(x)=1Γ(n)(1λ)nxn−1e−x1λSimplifying, we get:f(x)=λnΓ(n)xn−1e−λxTherefore, the sampling frame distribution of W=2λX¯ follows a gamma distribution with shape parameter n and scale parameter 2λ.b) To construct a 100(1−α)% 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~χ2n2To construct a confidence interval for λ, we can rearrange this equation:2nX¯λ2~χ2n2Next, we can find the critical values of the chi-squared distribution at the (1−α/2) and α/2 percentiles, denoted as χα/22 and χ1−α/22, respectively, with 2n degrees of freedom.The confidence interval for λ is then given by:(2nχ1−α/221X¯,2nχα/221X¯)This is the 100(1−α)% 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.

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

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