(a) A lamp has two bulbs, each of a type with average lifetime 1000 hours. Assuming that we can

(a) A lamp has two bulbs, each of a type with average...

Key Concepts

Exponential Distribution

The exponential distribution is a continuous probability distribution often used to model the time between events in a Poisson process, such as the lifetime of a component. It is characterized by its constant hazard rate and its mean, which is the reciprocal of the rate parameter. This distribution is fundamental in reliability analysis and survival studies.

Memoryless Property

The memoryless property is a key characteristic of the exponential distribution. It states that the probability of an event occurring in the future is independent of how much time has already elapsed. This property simplifies the analysis of sequential events or repairs, as past outcomes do not affect future probabilities.

Independence

Independence in probability theory refers to the situation where the occurrence of one event does not affect the probability of another event occurring. In reliability and failure analysis, assuming independent failures allows for the computation of joint probabilities by multiplying individual probabilities, which is important when evaluating systems with multiple components.

Convolution and Gamma Distribution

When considering the sum of two or more independent exponential random variables, the resulting distribution is a gamma distribution. Convolution is the process used to combine these independent random variables. The gamma distribution is crucial when dealing with the total time until a specified number of events occur, which often appears in problems related to sequential component failures or repairs.

Transcript

00:02 So you have a lamp with two light bulbs and then these light bulbs have a can be model with a it's a life, can be model with an exponential function.

00:32 So it would mean exponential probability function would mean a thousand hours.

00:41 So the probability that the light bulb fails at a time is gonna be p to the minus t over a thousand is that is a probability as distribution over a thousand this is the probability that the failure time is less than t so when it burns out turn this out so if we have a lamp with two light bulbs like this so what is the probability but both of the times t1 or t2 are less than a thousand hours so that would be so we will multiply the two the probability functions and then all those are into independent variables so we can do e to the minus t1 1 ,000 times e to the minus t 2 1 ,000 like this the t 1 d t2 and since the the bounds are independent of each other we can simply see this as a single interval squared the interval from 0 up to a thousand of e to the minus t over a thousand divided by a thousand t so well that squared so doing this all that has an antiterative e to the minus minus t over a thousand so that value it got 1 ,000 and 0, which all this number would be minus e to the minus 1, minus 0, minus 0, e to the 0, e to the 0 is 1.

03:14 So it is that, and then well this is the same as this is 1, minus e to the 1 over e, that is 2.

03:24 So that is the probability that the two light bulbs probability that the two light bulbs fell within the first 1 ,000 hours.

03:53 With all this have, they have mean light a thousand hours under that probability distribution.

04:03 So that is this probability.

04:05 Now assuming that we have one light bulb like that one and then there is another light bulb that fails this one fails less and less than a thousand hours mean t star less than a thousand hours also let's call this time see and then this light bulb is replaced by another new light bulb so this light bulb let's say that this bulb light has time white and it's get switched so what is the probability now that these two light bulbs the old one and this new light bulb that got replaced the probability that these two that the two fell in less less than an hour less than a thousand hours so for that what we have to do a condition and probability so it's gonna be the probability that well it's called this the light ball the time here for this one x the time for the new one y and the time for this old one z so this is the probability that x and y they are both less than a thousand hours so for this new time since y that we placed also we should write it like this the probability that x is less than a thousand and that y minus the time where that replaced that is less than an hour because a thousand because well this one get burned fails in a time less than a thousand so it is that probability given that uh c is less than a thousand so this is a joint probability joint probability and so well you can be model with the three variables.

07:10 So you have here the probability for this one to fail is the interval from 0 up to a thousand and then we have the distribution.

07:22 The probability the distribution is e to the minus x divided by a thousand and then that divided by a thousand.

07:29 Now this all the light bulb c also tells that distribution and then this new light bulb has the same distribution so you have to divide by a thousand divided by a thousand so those multiplied and then we integrate the y but uh since y has a shorter time, the time that y has to live, to meet the condition is a thousand minus the time for z...

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