High-speed Internet customers are oftrated by modem crashes....

High-speed Internet customers are oftrated by modem crashes. Suppose the time to "failure" for one particular brand of cable modem, measured in hundreds of hours, follows a gamma distribution with $\alpha=\beta=2$ . (a) Determine and graph the reliability function for this brand of cable modem. (b) What is the probability such a modem does not need to be refreshed for more than 300 $\mathrm{h} ?$ (c) Find the mean time to "failure" for such a modem. Verify that your answer matches the formula for the mean of a gamma rv given in Sect. 3.4. (d) Determine and graph the hazard function for this type of modem.

High-speed Internet customers are oftrated by modem crashes. Suppose the time to "failure" for one particular brand of cable modem, measured in hundreds of hours, follows a gamma distribution with $\alpha=\beta=2$ .
(a) Determine and graph the reliability function for this brand of cable modem.
(b) What is the probability such a modem does not need to be refreshed for more than 300 $\mathrm{h} ?$
(c) Find the mean time to "failure" for such a modem. Verify that your answer matches the formula for the mean of a gamma rv given in Sect. 3.4.
(d) Determine and graph the hazard function for this type of modem.

Key Concepts

Gamma Distribution

The gamma distribution is a two-parameter family of continuous probability distributions often used to model waiting times until a specified number of events occur. It is defined by a shape parameter (?) and a rate (or inverse scale) parameter (?), allowing for flexible modeling of skewed data and non-negative outcomes. Its probability density function and cumulative distribution function are commonly used in reliability analysis and other areas of applied probability.

Reliability Function

The reliability function, also known as the survival function, represents the probability that a system or component will operate without failure beyond a given time. Mathematically, it is defined as S(t) = 1 - F(t), where F(t) is the cumulative distribution function of the time-to-failure distribution. This function is central in reliability and life data analysis for determining system performance over time.

Mean Time to Failure

The mean time to failure (MTTF) is a key metric in reliability engineering that quantifies the expected operational lifetime of a system or component before a failure occurs. For a gamma distribution, the MTTF is calculated as the ratio of the shape parameter to the rate parameter (?/?). This measure helps in assessing the overall reliability and planning maintenance or replacement schedules.

Hazard Function

The hazard function, or failure rate, describes the instantaneous rate of failure at any given time, conditioned on survival up to that time. It is defined as h(t) = f(t)/S(t), where f(t) is the probability density function and S(t) is the reliability function. The hazard function is integral in understanding how the risk of failure evolves over time and in making strategic decisions based on component reliability.

Transcript

00:01 So for the a part, we know that pdf.

00:05 So pdf for the gamma distribution, for the gamma distribution is given by, so we have f of x.

00:21 So we equal to beta to the power alpha, all divided by gamma, gamma of alpha, times x to the power alpha, times x to the power alpha minus 1, times a to the power minus 1, beta x where we have alpha is greater than zero and beta two is greater than zero so from our question we know alpha and beta to be equal to two so we can substitute these values into our pdf so from here we are going to get two to the power two divided by gamma two i'm going to get alpha to the power two minus one times eight to the power minus 2 x.

01:22 So here we are going to get 4 and gamma 2, sorry, so we know gamma 2, gamma 2 is equal to 1.

01:46 So we are going to get 1 over here.

01:49 So we get x and we have e to the 4 minus 2x.

01:59 So from here we are going to get 4x times e to the power minus 2x.

02:10 X so this is the pdf for the gun so for the b part you are going to find probability of x less than or equal to 300 so from this we are going to get integral 0 to 300 and we know our pdf over here so f of x the x we know f of x to be 4x so we are going to get integral 0 to 300, 4x, e to the power negative to x.

02:53 So we can integrate from here.

02:58 So we can bring four out.

02:59 We have 0 to 300.

03:03 We have xe minus 2x the x.

03:24 So integrating this, you are going to get 4 times e to the power negative 2x.

03:33 So in here, we are going to get minus x over 2, minus 1 over 4.

03:44 And now interval is 0 to 300...

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