The probability density function of the time to failure of...

The probability density function of the time to failure of an electronic component in a copier (in hours) is $f(x)=$ $e^{-x / 1000 / 1000}$ for $x>0 .$ Determine the probability that (a) A component lasts more than 3000 hours before failure. (b) A component fails in the interval from 1000 to 2000 hours. (c) A component fails before 1000 hours. (d) The number of hours at which $10 \%$ of all components have failed

The probability density function of the time to failure of an electronic component in a copier (in hours) is $f(x)=$ $e^{-x / 1000 / 1000}$ for $x>0 .$ Determine the probability that
(a) A component lasts more than 3000 hours before failure.
(b) A component fails in the interval from 1000 to 2000 hours.
(c) A component fails before 1000 hours.
(d) The number of hours at which $10 \%$ of all components have failed

Key Concepts

Exponential Distribution

The exponential distribution is a continuous probability distribution frequently used to model the time between independent events that occur at a constant average rate, such as the time until failure of a component. It is defined by its rate parameter (often denoted by ?, where ? is the inverse of the mean) and is characterized by the memoryless property, meaning that the future probability of an event is independent of past events.

Survival Function

The survival function, also referred to as the reliability function, provides the probability that the time to event (or failure) exceeds a certain value. It is calculated for the exponential distribution as the complement of the cumulative distribution function (CDF), typically written as S(x) = exp(–?x). This function is used to determine the probability that a component lasts more than a given time.

Cumulative Distribution Function (CDF)

The cumulative distribution function of a probability distribution gives the probability that the random variable takes on a value less than or equal to a specified value. In the context of the exponential distribution, the CDF is expressed as F(x) = 1 – exp(–?x), which facilitates the calculation of the probability that a failure occurs within a specific time interval.

Quantile Function

The quantile function is essentially the inverse of the CDF and is used to determine the value below which a certain percentage of observations fall. For the exponential distribution, solving 1 – exp(–?x) = p for x yields the time corresponding to the p-th percentile. This is particularly useful in determining thresholds such as the time by which 10% of components are expected to have failed.

Transcript

00:01 Hey guys in this problem we have the probability mass function f of x it's 1 over 1 ,000 times e power negative x over 1 ,000 okay for x more than 0.

00:17 So let's find the probability of x more than 3 ,000 so it's integration from 3 000 to infinity for 1 over 1 ,000 times e power negative x over 1 ,000 d x.

00:37 Okay, so this is 1 over 1 ,000 times e power negative x over 1 ,000.

00:51 Okay, over negative 1 over 1 ,000.

00:57 And then we should substitute by the limits infinity and 3 ,000.

01:03 So this is 3 zeros, okay? okay, so this is negative 1 times e power negative infinity minus e power negative 3.

01:18 Okay, so it's e power negative 3, which is 0 .0498.

01:28 Okay, now let's find the probability of x, where x more than 1 ,000, i'm sorry.

01:37 Okay, 1000 less than x and less than to a thousand.

01:45 Okay, so it's integration from 1 ,000 to 2 ,000 for 1 over 1 ,000 e power negative x over 1 ,000 dx.

01:59 Okay, okay, so this is 1 over 1 ,000 times 1 ,000.

02:10 E for negative x over 1 ,000 over negative 1 over 1 ,000.

02:18 Then we should substitute by the limits to 1000 and 1 ,000.

02:25 So it's e for negative 1 minus e for negative 2.

02:31 So this probability is 0 .236.

02:37 Okay.

02:38 Okay.

02:39 Then we need to find probability of x where x less than 1 ,000 okay so it's 1 over 1 ,000 times integration from 0 to 1 ,000 for e per negative x over 1 ,000 d x okay so this is 1 over 1 ,000 times e power negative x over 1 ,000 times e negative 1 over 1000.

03:15 Okay, okay so this is by the limits which is 1000 and 0...

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