To determine the type of lightbulb in the box, we can perform a hypothesis test. The null hypothesis is that the box contains type-A lightbulbs, and the alternative hypothesis is that the box contains type-B lightbulbs.
To test the null hypothesis, we can collect a sample of lightbulbs from the box and measure their lifetimes. We can then use the sample mean and standard deviation to calculate the test statistic, which is the z-score of the sample mean. If the z-score is significantly different from zero (i.e. outside of the critical region), we can reject the null hypothesis and conclude that the box contains type-B lightbulbs.
Alternatively, we can use the Bayesian approach and update our prior probability of the box containing type-B lightbulbs based on the observed lifetimes of the lightbulbs in the sample. If the posterior probability is greater than 1/4, we can conclude that the box contains type-B lightbulbs.
Problem 1. Determining the type of a lightbulb.
The lifetime of a type-A bulb is exponentially distributed with parameter 𝜆 . The lifetime of a type-B bulb is exponentially distributed with parameter 𝜇 , where 𝜇>𝜆>0 . You have a box full of lightbulbs of the same type, and you would like to know whether they are of type A or B. Assume an a priori probability of 1/4 that the box contains type-B lightbulbs.
This question was a pain in the arse, so I am posting its solution here for posterity.
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1. ln(mu/(3*lambda))/(mu-lambda)
2. (a): (1/4)*e^(-mu*alpha) + (3/4)(1-e^(-lambda*alpha))
3. 0.3286
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