I was wondering if anyone can check my logic here. I'm given light bulbs that are used 24/7 and each light blubs will last on average 25 days. The light bulbs come in a pack of 12. The question is find the probability that the pack of light bulbs will last more than a year.
What I've done:
Given $N$ = sample size, which is 12. $\mu_{ind} = 25$
$\mu = N*\mu_{ind} = 12 * 25 = 300$
$\delta = \sqrt{N} * 25 = 86.60$
$z = (x-\mu)/\delta = (365 - 300)/86.60 = 0.7505$
using the z-table i got $0.2734$ then subtract that with $0.5$ because we want to find "more than" 365 days which gives me $0.2266$ or $22.66$%.
We're suppose to find the exponential distribution and find the central limit theorem but I did it this way. I was wondering if this apporach was correct.
Hint: If the wait time between events follows an exponential distribution, then the count of events within an interval follows a Poison distribution.
Hint 2: A Poisson distribution is approximately Gaussian/Normal (some more so than others). (CLT)
Hint 3: A Poisson random variable is discrete over Integer-values. Use Continuity Correction. $$\mathsf P(Y\geq y)=\mathsf P(Y> y-\tfrac 12)$$
Hint 4: That looks like what you did. So...