# Question about Probability of operating time of transistors (MTBF)

• math4everyone
In summary, the conversation discusses the use of exponential distributions to calculate probabilities in a variety of scenarios, including the joint probability of two events occurring, the probability of events occurring within a specific time frame, and the concept of memorylessness in exponential distributions. The solution involves using exponential CDFs and their complements to calculate the desired probabilities.
math4everyone

## Homework Equations

$$f_X(x)=\lambda e^{-\lambda x}$$
$$F_X(x) = 1-e^{-\lambda x}$$
$$\mu = \frac{1}{\lambda}$$

## The Attempt at a Solution

a) $$f_{X,Y}(x,y) = f_X(x)f_Y(y) = \frac{1}{800} e^{-\frac{1}{800}x} \frac{1}{1000}e^{-\frac{1}{1000}y}$$
$$=\frac{1}{800000}e^{-\left(\frac{1}{800}x+\frac{1}{1000}y\right)}$$
$$F_{X,Y}(x,y) = F_X(x)F_Y(y)=(1-e^{-\frac{1}{800}x})(1-e^{-\frac{1}{1000}y})$$
$$F_{X,Y}(1000,1000) = (1-e^{-\frac{1000}{800}})(1-e^{-\frac{1000}{1000}})=0.451$$
b) I am having trouble with this
c) Apply the same concept of a) and get:
$$F_{X,Y}(1000,1000,1000) = (1-e^{-\frac{1000}{800}})(1-e^{-\frac{1000}{1000}})(1-e^{-\frac{1000}{500}})=0.389$$
d) I am also struggling with this
$$P(\text{won't fail in the next 1000 hours}|\text{didn't fail in the first 1000 hours})$$
$$=\frac{P(\text{won't fail in the next 1000 hours and didn't fail in the first 1000 hours})}{P(\text{didn't fail the first 1000 hours})}$$
$$=\frac{P(\text{won't fail in the next 1000 hours and didn't fail in the first 1000 hours})}{1-F_X(1000,1000,1000)}$$
$$=\frac{P(\text{won't fail in the next 1000 hours and didn't fail in the first 1000 hours})}{0.61}$$
I am having trouble of finding the probability of the numerator... Is it $$(1-F_X(1000,1000,1000))^2$$?

math4everyone said:
b) I am having trouble with this
Suppose the first fails after time t. What is the probability density for that? How long does the second one have for both to have failed in 1000h? What is the probability density of the overall event?

If you are familiar with Poisson counting processes -- I'd suggest using that for B -- You basically want the complement to ___.

Using exponential CDFs and their complements should be get you there for the other problems.

Hint: recall that that exponential distributions exhibit memorylessness. This should be immensely helpful in one case and is important to remember in general.

## 1. How is the probability of operating time of transistors calculated?

The probability of operating time of transistors is typically calculated using a statistical method known as the Mean Time Between Failures (MTBF). This involves collecting data on the number of failures and the total operating time of a large sample of transistors, and then using that data to estimate the average time between failures.

## 2. What factors influence the probability of operating time of transistors?

There are several factors that can influence the probability of operating time of transistors, including the quality of the manufacturing process, the environmental conditions in which the transistors are used, and the stress placed on the transistors during operation. Other factors such as voltage fluctuations and temperature changes can also impact the probability of operating time.

## 3. Can the probability of operating time of transistors be improved?

Yes, the probability of operating time of transistors can be improved by using higher quality materials and manufacturing processes, implementing proper testing and quality control measures, and optimizing the operating conditions and stress levels for the transistors. Regular maintenance and monitoring can also help to improve the probability of operating time.

## 4. How does the probability of operating time of transistors affect overall system reliability?

The probability of operating time of transistors is directly related to the overall reliability of a system. If the probability of operating time is low, it means that the transistors are less likely to fail, resulting in a more reliable system. On the other hand, if the probability of operating time is high, it increases the chances of transistor failure, which can lead to system failures and downtime.

## 5. Are there any techniques for predicting the probability of operating time of transistors?

Yes, there are various techniques and models that can be used to predict the probability of operating time of transistors. These include the MTBF method mentioned earlier, as well as other methods such as Weibull analysis and reliability growth analysis. These techniques use statistical and mathematical tools to analyze data and make predictions about the probability of operating time.

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