(a) A type of light bulb is labeled as having an average lifetime of 1000 hours. It's reasonable

(a) A type of light bulb is labeled as having an average...

Key Concepts

Median of the Exponential Distribution

The median of the exponential distribution is the time by which 50% of the population is expected to have experienced the event (e.g., failure). It is derived by setting the survival function equal to 0.5, leading to the formula median = (ln 2)/?. This measure is a useful summary statistic that provides insight into the central tendency of lifetimes modeled by the exponential distribution.

Cumulative Distribution Function (CDF) and Survival Function

The cumulative distribution function (CDF) for an exponential distribution is F(x) = 1 – e^(–?x), which gives the probability that the event occurs by time x. Conversely, the survival function S(x) = e^(–?x) represents the probability that the event has not occurred by time x. These functions are critical for determining probabilities of failure or survival over given time intervals.

Exponential Distribution

The exponential distribution is a continuous probability distribution often used to model the time between events in a Poisson process. It is characterized by a constant hazard rate and is widely utilized in reliability analysis and lifetime modeling. Its memoryless property signifies that the future lifetime is independent of the elapsed time, making it a natural choice for modeling the time until failure of devices.

Probability Density Function (PDF)

The probability density function (PDF) for an exponential distribution is given by f(x) = ?e^(–?x) for x ? 0, where ? is the rate parameter. This function describes the likelihood of the random variable taking on a specific value and is fundamental in calculating probabilities over specific intervals of time.

Transcript

00:07 Okay, so for this question, we're trying to model the behavior of light bulb, like when it is going to fail.

00:21 And you're given the average, every time for a light bulb to fail, is 1 ,000.

00:28 So the average is mu, is equals 1 ,000 hours.

00:34 So that's how long a light valve is supposed to last for before it fails.

00:38 And here we're supposed to use, so we're supposed to use the probability dense function as an exponential which is defined like this, one over mule, e to minus one of a mule t.

01:03 That is if t is greater than all equals zero, otherwise f of t will be zero for all t less than zero.

01:18 Now so now for roman what is the probability that a lipo lasts for 200 hours so to handle this question is the same as doing what is p of t where t is a random variable is less than 200 right and then translating this into an integral it becomes the integral from 0 to 200 and then you have so f one of a meal is one of 1 ,000 so 1 over 1000 to minus 1 of a mu which is minus 1 over 1 ,000 t d t i'm only integrating for t greater than o equals 0 because there f of t is non -zero and f of t is zero equal else so now create myself some room here so basically now evaluating this integral so the 1 over 1 ,000 is a constant so that will come outside so you get one over 1 ,000 from 0 to 200 and then you have e to the minus 1 of 1 ,000 of t d t so evaluating this integral i will obtain 1 ,000 and then the integral itself becomes you get e to the minus 1 over 1 ,000 t and then divided by the coefficient in front of t, that means i'll divide by minus 1 over 1 ,000.

03:48 And then i have to evaluate from 0 to 200.

04:04 Now, you can clearly see that these 1 ,000 and this one will cancel, but you will obtain a minus sign there.

04:16 So that means you will get e to the minus 1 over 1 ,000.

04:25 This is supposed to be 0 multiply by t so the apple limit is 200 so i'll put 200 here well there's a minus sign here too so and then plus so minus minus will come a plus so e to the zero when i plug in t equals 0 so i go 1 so which is equals 2 now some zeros are going to cancel here so these two zeros and these two zeros will cancel i cancel these two i go one here i cancel two here i can't five so that means i will obtain minus the exponential of minus one over five plus one which is roughly equals to zero point it's minus 0 .81 87 plus one so so which roughly equals to 0 .18, 13%.

06:06 So if you can root that into a percentage.

06:09 So these are the answers we're looking for.

06:14 So that's the percentage of light bulbs that are going to fail within the first 200 hours.

06:24 Now moving on to b, so roman 2, not b, so roman 2, what's the probability that, you know, a life of lasts for more than 800 hours? so this is the same as asking what is p of t greater than 800? so that's the question we try and answer here.

06:52 So which becomes this is the same as doing the integral from 800 to infinity, and the f of t here, dt.

07:03 So that's the integral we need to worry about.

07:12 So i'll open a new tab here.

07:24 So now doing this integral, so we do from, we're trying to answer the question what is p of t is greater than 800.

07:36 So which i said earlier this is equivalent to integrating from 800 to infinity, 1 over 1 ,000, e to minus 1 over.

08:05 So this integral is equivalent here.

08:11 Then we change colors here.

08:13 So this is, i'll change this into a limit.

08:17 So which will become, i'll push the one of one thousand outside.

08:23 So then i'll change this into limit.

08:30 So the reason why i'm doing this is because i don't like dealing with infinity here.

08:35 So i'm going to change this infinity to x like this.

08:39 So i'll be integrating from 800 to x and then e to the minus 1 over 1000 td td and then that after i can take the limit.

08:59 So integrating we obtain one of 1000.

09:07 Once i integrate this part i will obtain.

09:10 I still have my limit here.

09:12 So limit x going to infinity.

09:20 So i got e to 3 ,000.

09:22 The minus 1 over 1000 t divided by the coefficient in front of t which is minus 1 over 1 ,000 and then of course you evaluate this from 800 to x doing this you can clearly see that there's 1 ,000 and this 1 ,000 one of 1 ,000 will cancel but you'll gain this minus sign here so so i'm going to create myself some room here.

10:16 Now, so this is equivalent to, so a minus, and then you got the limit, x going to infinity.

10:39 Now, if i plugging t equals x, i get e to the minus 1 over 1 .000...

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