1. ## sample paths

<H2>Suppose that the lifetime of a light bulb is exponentially distributed and that the expected lifetime is 250 hours.
Suppose that we switch on the light bulb at midnight and then we check at midnight every day afterwards in order to see if the light bulb is still burning.
Let Xn=1 if the bulb is still burning and Xn=0 if it has burnt out, where n=1,2,3,… is the number of days after we start the light bulb.

Sketch two sample paths of for n=1 to 10

</H2>

2. Originally Posted by winganger
<H2>Suppose that the lifetime of a light bulb is exponentially distributed and that the expected lifetime is 250 hours.
Suppose that we switch on the light bulb at midnight and then we check at midnight every day afterwards in order to see if the light bulb is still burning.
Let Xn=1 if the bulb is still burning and Xn=0 if it has burnt out, where n=1,2,3,… is the number of days after we start the light bulb.

Sketch two sample paths of for n=1 to 10

</H2>
There are only two states that the bulb can be in 1 it is still burning and 0 it is burnt out.

It starts at 1 and remains at 1 untill it burns out when it becomes 0, and then remains at 0.

So any sample path consists of a number of 1 followed by 0's. The number of 1's could be any number from zero to ten, as could the number of 0's (well 10 minus the number of 1's).

RonL

3. Originally Posted by CaptainBlack
There are only two states that the bulb can be in 1 it is still burning and 0 it is burnt out.

It starts at 1 and remains at 1 untill it burns out when it becomes 0, and then remains at 0.

So any sample path consists of a number of 1 followed by 0's. The number of 1's could be any number from zero to ten, as could the number of 0's (well 10 minus the number of 1's).

RonL
Thank you very much.
I would like to ask one more question,
how to find the mean of Xn, for n=0,1,2,...

4. Originally Posted by winganger
Thank you very much.
I would like to ask one more question,
how to find the mean of Xn, for n=0,1,2,...
Won't it just be

$\displaystyle 1 \cdot \Pr(Y \geq n) + 0 \cdot \Pr(Y < n) = .....$,

where Y is the random variable lifetime of bulb (days) .....