To determine the maximum likelihood estimate (MLE) for the hazard rate $\lambda$ of the exponential distribution given the failure times, we can follow these steps:

Step 1: Understand the Problem

The failure times are assumed to follow an Exponential distribution with hazard rate $\lambda$. The probability density function (PDF) for an exponential distribution is given by:

$f(t|\lambda) = \lambda e^{-\lambda t}$

Given the dataTo determine the maximum likelihood estimate (MLE) for the hazard rate $\lambda$ of the exponential distribution given the failure times, we can follow these steps:

Step 1: Understand the Problem

The failure times are assumed to follow an Exponential distribution with hazard rate $\lambda$. The probability density function (PDF) for an exponential distribution is given by:

$f(t|\lambda) = \lambda e^{-\lambda t}$

Given the

$\text{Failure times: } 6.8, 6.7, 6.9, 0.7, 5.6, 3.8, 6.1, 3.8, 1.8, 4.1$

Step 2: Write the Likelihood Function

The likelihood function for the given data is the product of the individual densities:

$L(\lambda) = \prod_{i=1}^{n} \lambda e^{-\lambda t_i}$

where $t_i$ are the observed failure times and $n$ is the total number of observations (here $n = 10$).

Step 3: Simplify the Likelihood Function

The likelihood function simplifies to:

$L(\lambda) = \lambda^n e^{-\lambda \sum_{i=1}^{n} t_i}$

Step 4: Take the Log-Likelihood

Taking the natural logarithm of the likelihood function gives the log-likelihood function:

$\ln L(\lambda) = n \ln \lambda - \lambda \sum_{i=1}^{n} t_i$

Step 5: Differentiate and Solve for $\lambda$

To find the MLE, we differentiate the log-likelihood with respect to $\lambda$ and set it to zero:

$\frac{d}{d\lambda} \ln L(\lambda) = \frac{n}{\lambda} - \sum_{i=1}^{n} t_i = 0$

Solving for $\lambda$:

$\hat{\lambda} = \frac{n}{\sum_{i=1}^{n} t_i}$

Step 6: Calculate the MLE

Substitute the given failure times into the formula:

$\hat{\lambda} = \frac{10}{6.8 + 6.7 + 6.9 + 0.7 + 5.6 + 3.8 + 6.1 + 3.8 + 1.8 + 4.1}$

Now, let's calculate the sum of the failure times and then the MLE for $\lambda$.

$\sum_{i=1}^{n} t_i = 46.3$

Thus:

$\hat{\lambda} = \frac{10}{46.3} \approx 0.216$

Final Answer

The maximum likelihood estimate (MLE) for $\lambda$ is:

$\hat{\lambda} \approx 0.216 \text{ (to three decimal places)}$

Would you like more details on any part of this solution, or have any further questions?

Related Questions:

1. What is the significance of the hazard rate $\lambda$ in the context of an exponential distribution?
2. How does the assumption of independent and identically distributed (iid) failures influence the MLE calculation?
3. What are some real-world applications of estimating the hazard rate for components?
4. How would the MLE for $\lambda$ change if one additional component was observed without failure?
5. How does the exponential distribution compare to other distributions commonly used in survival analysis?

Tip: When dealing with exponential distributions in reliability studies, the hazard rate $\lambda$ is the reciprocal of the mean time to failure.