Estimation Theory

Step 1: Understanding the Concept:
The given cumulative distribution function (CDF) is that of an exponential distribution. The problem asks for the Maximum Likelihood Estimator (MLE) of the expected value, E(X). We will use the invariance property of MLEs, which states that if is the MLE of , then the MLE of a function is .

Step 2: Key Formula or Approach:
1. Identify the distribution and its parameter(s) from the CDF. The PDF is . 2. Find the expected value, E(X), in terms of the parameter . 3. Find the MLE of the parameter , denoted . 4. Apply the invariance property to find the MLE of E(X).

Step 3: Detailed Explanation:
1. The CDF is for . The corresponding PDF is . This is an exponential distribution with rate parameter . 2. The expected value (mean) of an exponential distribution with rate is . 3. To find the MLE of , we first write the likelihood function for a sample : The log-likelihood function is: Differentiate with respect to and set to zero: Solving for , we get the MLE of : 4. Using the invariance property of MLEs, the MLE of is:
Step 4: Final Answer:
The MLE of E(X) is the sample mean, .

Step 1: Understanding the Concept:
The problem requires finding the expected value of the reciprocal of a statistic , where is the sum of transformed random variables. The key is to first determine the probability distribution of and then use the definition of expected value.

Step 2: Key Formula or Approach:
1. Find the distribution of a single transformed variable, , using the change of variable technique for probability distributions. 2. Recognize that is the sum of i.i.d. random variables. The distribution of a sum of certain i.i.d. variables (like Exponential) is a known distribution (Gamma). 3. Calculate by integrating over the support of Y.

Step 3: Detailed Explanation:
Let's find the distribution of . The CDF of is for . The CDF of for is: . This is the CDF of an Exponential distribution with rate parameter . So, are i.i.d. Exponential( ) random variables. . The sum of i.i.d. Exponential( ) variables follows a Gamma distribution with shape parameter and rate parameter . Thus, . The PDF of is for , with . Now, we calculate : The integral is related to the Gamma function. The Gamma function is defined as . Using the substitution : Substitute this back into the expectation formula: Using the property : Since , the final result is:
Step 4: Final Answer:
The expected value is .

Step 1: Understanding the Concept:
The Maximum Likelihood Estimator (MLE) of a parameter is the value of the parameter that maximizes the likelihood function for a given sample of data. The standard procedure is to write the likelihood function, take its natural logarithm (log-likelihood), differentiate with respect to the parameter, set the derivative to zero, and solve for the parameter.

Step 2: Key Formula or Approach:
1. Write the likelihood function: . 2. Write the log-likelihood function: . 3. Solve the equation: for .

Step 3: Detailed Explanation:
Given a random sample : 1. The likelihood function is: 2. The log-likelihood function is: 3. Differentiate with respect to : 4. Set the derivative to zero and solve for the MLE, :
Step 4: Final Answer:
The MLE of is .

Step 1: Understanding the Concept:
To find the confidence limits for a population correlation coefficient ( ), we use Fisher's Z-transformation. This method transforms the skewed sampling distribution of into an approximately normal distribution, for which a standard confidence interval can be constructed. The limits of this interval are then transformed back to the original correlation scale.

Step 2: Key Formula or Approach:
1. Transform the sample correlation to : . 2. Calculate the standard error of : . 3. Construct the confidence interval for the transformed correlation: . 4. Transform the interval limits back to the scale using the inverse function: .
Step 3: Detailed Explanation:
Given values: - Sample size, .
- Sample correlation, .
The confidence level is not specified. The term "correlation limits" often implies a high-confidence range. Let's try constructing a 99.7% confidence interval (which corresponds to ), as this is a common interpretation for "limits".
1. Fisher's Z-transformation: 2. Standard Error: 3. Confidence Interval for (using Z=3): - Lower z-limit: - Upper z-limit: 4. Inverse Transformation: - Lower r-limit: - Upper r-limit: The resulting limits are approximately (0.771, 0.825), which closely matches option (B).
Step 4: Final Answer:
The correlation limits for the entire population are approximately (0.773, 0.827).

Step 1: Understanding the Concept:
A sufficient statistic for a set of parameters is a function of the sample data that captures all the information about the parameters contained in the sample. The Fisher-Neyman Factorization Theorem is the standard tool to find a sufficient statistic.

Step 2: Key Formula or Approach:
The Fisher-Neyman Factorization Theorem states that a statistic is sufficient for if and only if the joint probability density function can be factored into two functions: where depends on the data only through the statistic , and does not depend on the parameter .

Step 3: Detailed Explanation:
The probability density function (PDF) for a Beta distribution of the first kind is: For a random sample of size , , the joint PDF (or likelihood function) is the product of the individual PDFs: Let's identify the parts for the factorization theorem. Let . Let's call the components and . Then we can write the likelihood as: Here, - . This function depends on the data only through the statistic . - . This function does not depend on the parameters or . Since the joint PDF can be factored in this way, by the Fisher-Neyman Factorization Theorem, the statistic is a sufficient statistic for .
Step 4: Final Answer:
The sufficient statistic for is .

Step 1: Understanding the Concept:
The problem asks for the variance of the Maximum Likelihood Estimator (MLE) of the parameter of a Poisson distribution. The first step is to find the MLE, , and the second step is to calculate its variance.

Step 2: Key Formula or Approach:
1. Find the MLE of , which is a standard result ( ). 2. Find the variance of the MLE using properties of variance. 3. Since the are independent, . 4. For a Poisson( ) distribution, .

Step 3: Detailed Explanation:
Part 1: Finding the MLE The likelihood function for a sample is: The log-likelihood is: Differentiate with respect to and set to zero: So, the MLE is . Part 2: Finding the Variance of We need to find . Using the variance property : Since the observations are from a random sample, they are independent. Thus, the variance of the sum is the sum of the variances: For a Poisson( ) distribution, the variance of a single observation is .
Step 4: Final Answer:
The variance of the MLE is .

Step 1: Understanding the Concept:
The question asks for a 99% confidence interval for the population mean height. Since the population standard deviation ( ) is known and the sample size is large (n>30), we can use the Z-distribution to construct the interval.

Step 2: Key Formula or Approach:
The formula for a confidence interval for the population mean ( ) when is known is: Where: - is the sample mean.
- is the population standard deviation.
- is the sample size.
- is the critical Z-value for the desired confidence level.

Step 3: Detailed Explanation:
First, identify the given values: - Sample size, .
- Sample mean, inches.
- Population standard deviation, inches.
- Confidence level = 99%, which means .
Next, find the critical Z-value, . - .
- We need the Z-value such that the area in the upper tail of the standard normal distribution is 0.005.
- From the Z-table or standard statistical knowledge, .
Now, calculate the margin of error (ME): Finally, construct the confidence interval: - Lower limit =
- Upper limit =
The confidence interval is approximately (63.2, 64.8).

Step 4: Final Answer:
The 99% confidence limits for the mean population height of plants are (63.2, 64.8).

Step 1: Understanding the Concept:
The likelihood function, , for a given set of observations is the product of the probability density function (PDF) evaluated at each of these points. We need to calculate this value for the specific parameter value .

Step 2: Key Formula or Approach:
The likelihood function is given by: We need to calculate .

Step 3: Detailed Explanation:
First, we evaluate the PDF at for each observation. The PDF at is: Now, we find the value of the PDF for each observation: - For : Since , we use the first case. - For : Since , we use the second case. - For : The value is the boundary point and is included in both intervals. The function is continuous at , so either formula gives the same result. Let's use the first one. Now, we compute the likelihood by multiplying these values:
Step 4: Final Answer:
The value of the likelihood function at is .