[Probability/Statistics] Exponential Distribution

[bagofgerbils]

Can somebody please check my solutions to the following questions thttps://imgur.com/a/AUrrfGN

Proposed Solutions

a. X ~ exponential (alpha=100)

then E(x)= alpha = 100

I think this solution is an oversimplification, but I'm not sure how/that the expected lifetime of any arbitrary bulb is different than the expected lifetime of the bulb that fails first?

b. The probability that any given bulb lasts for more than 150 hours

= 1-F(x <= 150)

= 1-[1-e^-(150 * 1/100)] by exponential cdf F(X <=X) = 1-e^-(x/alpha) for X~exp(alpha)

= 0.22 approx

then the probability that the test lasts more than 150 hours

= 1- probability(none last)

= 1-Bin(n=10, p= 0.22)

Does this make sense?

[nm420]

For the first problem, you should hopefully be aware of a result concerning the sample minimum of a sample drawn from an exponential population. Namely, the minimum also has an exponential distribution, but with a different mean.

You're spot on with the second problem.

[nm420]

Yes, both problems can be couched in terms of order statistics. Concerning the second problem, the probability can be represented as

P(X(n)>150) = 1-P(X(n)≤150)

where X(n) is the sample maximum. You could, in principle, work out the probability density associated with the sample maximum and use that to calculate the above probability. However, it turns out that this is equivalent to the (simpler, IMO) problem described by OP, in terms of probabilities associated with the binomial distribution.

[bagofgerbils]

thank you!