- #1

Mayan Fung

- 131

- 14

My friend is now taking an introductory course about statistics. The professor raised the following question:

A light bulb has a lifespan with a uniform distribution from 0 to 2/3 years (i.e. with a mean of 1/3 years). You change a light bulb when it burns. How many light bulbs are expected to burn in 2 years?

Intuition tells us the answer is 6. However, the professor said that the answer is smaller than 6. I tried a simulation with python and found that the value is around 5.6-5.7. The professor didn't provide any rigorous proof.

Here's my thought:

Let ##Y## be the number of light bulbs burnt in 2 years

Let ##x_i## be the lifespan of the ##i-th## light bulb

My goal is to find ##P(Y=n)##, i.e. the probability distribution, then find its expectation ##E(Y)##

Exactly ##n## bulbs burnt in 2 years = lifespan sum of first ##n## bulbs <2

##P(Y=n) = P( \sum_{i=1}^{n+1} x_i > 2 \cap \sum_{i=1}^n x_i < 2)##

I cannot proceed as I don't know how to evaluate the probability. I feel like that the question is about Markov chain in stochastic process but I am not sure.

A light bulb has a lifespan with a uniform distribution from 0 to 2/3 years (i.e. with a mean of 1/3 years). You change a light bulb when it burns. How many light bulbs are expected to burn in 2 years?

Intuition tells us the answer is 6. However, the professor said that the answer is smaller than 6. I tried a simulation with python and found that the value is around 5.6-5.7. The professor didn't provide any rigorous proof.

Here's my thought:

Let ##Y## be the number of light bulbs burnt in 2 years

Let ##x_i## be the lifespan of the ##i-th## light bulb

My goal is to find ##P(Y=n)##, i.e. the probability distribution, then find its expectation ##E(Y)##

Exactly ##n## bulbs burnt in 2 years = lifespan sum of first ##n## bulbs <2

**AND**lifespan sum of first ##n+1## bulbs > 2, i.e.##P(Y=n) = P( \sum_{i=1}^{n+1} x_i > 2 \cap \sum_{i=1}^n x_i < 2)##

I cannot proceed as I don't know how to evaluate the probability. I feel like that the question is about Markov chain in stochastic process but I am not sure.

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