Exponential Confidence Interval. It is known that, for large n: sqrtn(lambda barx-1) ∼ Normal (0,1)

ymochelows 2022-09-14 Answered
Exponential Confidence Interval
It is known that, for large n:
n ( λ x ¯ 1 ) N o r m a l ( 0 , 1 )
With this approximation, show that the 95% confidence interval for λ is:
n 1.96 n x ¯ , n + 1.96 n x ¯
I think I need to manipulate the formula for confidence intervals for exponential distributions but I'm not sure where to start (simplified version since large n?)
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Answers (1)

Adrienne Harper
Answered 2022-09-15 Author has 14 answers
Step 1
About 95% of the probability distribution of a standard normal distribution lies in the interval (-1.96, 1.96).
Step 2
So all you need to do is manipulate
P ( 1.96 < n ( λ x ¯ 1 ) < 1.96 ) = 0.95
to something like
P ( n 1.96 n x ¯ < λ < n + 1.96 n x ¯ ) = 0.95
remembering that x ¯ is a random variable while λ is not.

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