(a) A lamp has two bulbs of a type with an average lifetime...

(a) A lamp has two bulbs of a type with an average lifetime of 1000 hours. Assuming that we can model the probability of failure of these bulbs by an exponential density function with mean $\mu=1000,$ find the probontity that both of the lamp's bulbs fail within 1000 hours. (b) Another lamp has just one bulb of the same type as in part (a). If one bulb burns out and is replaced by a bulb of the same type, find the probability that the two bulbs fail within a total of 1000 hours.

(a) A lamp has two bulbs of a type with an average lifetime
of 1000 hours. Assuming that we can model the probability of failure of these bulbs by an exponential density
function with mean $\mu=1000,$ find the probontity that
both of the lamp's bulbs fail within 1000 hours.
(b) Another lamp has just one bulb of the same type as in
part (a). If one bulb burns out and is replaced by a bulb
of the same type, find the probability that the two bulbs
fail within a total of 1000 hours.

Key Concepts

Exponential Distribution

The exponential distribution is often used to model the time between events in a process where events occur continuously and independently at a constant average rate. It is characterized by its mean, and its probability density function decreases exponentially over time, making it a natural model for lifetimes in reliability studies.

Memoryless Property

A fundamental property of the exponential distribution is that it is memoryless; that is, the probability of an event occurring in the future is independent of the past. This means that the chance of failure in the next interval remains the same regardless of how long the component has already been operational.

Independent Events

When dealing with multiple components such as bulbs, independence implies that the failure of one component does not affect the probability of failure of another. This enables the use of multiplication for joint probability calculations, which is essential when assessing the likelihood of simultaneous or sequential failures.

Gamma Distribution

The gamma distribution is useful for modeling the sum of independent exponential random variables. In reliability problems, it is often applied to represent the total time until multiple failures occur, as it generalizes the exponential distribution to scenarios where more than one event (failure) is of interest.

Transcript

00:01 So for the first part, we are seeing the probability density function for bob 1, which is f of x, it's equal to this, when x is greater than or equal to 0.

00:13 And that of bob 2 would be this function, when y is greater than or equal to 0.

00:21 So the probability for interval of x, 0 ,000, then that of y, 0 ,000, is going to be the double integer.

00:31 Over the region r of a function d r, which is equal to so this then it's equal to the integral the integral to 0 ,000, you have 0 to 1000, you have the function, then, then is this and that, so you have e to the power.

01:04 Negative x divided by 1000 1000 then e to the power negative y divided by 1 ,000 divided by 1000 divided by 1000 1 ,000 squared the y the x so this will be so you have this day it's equal to you have the integral from 0 to 1 ,000, you have e exponent minus x divided by 1 ,000, divided by 1 ,000, the x.

01:55 Then the integral, 0 to 1 ,000, there are 0 to 1 ,000, e exponent negative y, divided by 1 ,000, divided by 1 ,000, y so this will be equal to i have e negative e exponents minus x divided by 1000 the interval zero to 1 ,000 then negative e exponents negative y divided by 1 ,000 the interval 0 to 1 ,000 and this will be equal to 1 minus 1 divided by e exponent 2, exponent 2 as the answer.

02:59 So what it means is that giving that the probability of failure of bugs can be modeled.

03:10 So the probability, so this is our results, 1 minus this.

03:21 So which is the final answer so if you are looking at the exponential if it was the exponential density function so let's say for exponential exponential density density function then what it means is that our f of x it's equal to e to the power negative x divided by mu divided by mu when x is greater than or equal to zero then it is zero when x is less than zero where mu here the mu here is the mean of the probability function probability function so this is the you have when giving the probabilities of failure of both.

04:38 For the second part, we are saying that the probability of density function for bob 1, it's f of x and that's above 2, it's f of y...

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