The interarrival time of queries at a call center are...

The interarrival time of queries at a call center are exponential random variables with mean interarrival time $1 / 4$. Suppose that a sample of size 9 is obtained. (a) The estimator $\hat{\lambda}_1=1 / \bar{X}_9$ is used to estimate the arrival rate. Find the probability that the estimator differs from the true arrival rate by more than 1 . (b) Suppose the estimator $\hat{\lambda}_2=1 / 9 \min \left(X_1, \ldots, X_9\right)$ is used to estimate the arrival rate. Find the probability that the estimator differs from the true arrival rate by more than 1. (c) Generate 100 random samples of size 9. Compare the probabilities obtained in parts a and b to the observed relative frequencies.

The interarrival time of queries at a call center are exponential random variables with mean interarrival time $1 / 4$. Suppose that a sample of size 9 is obtained.
(a) The estimator $\hat{\lambda}_1=1 / \bar{X}_9$ is used to estimate the arrival rate. Find the probability that the estimator differs from the true arrival rate by more than 1 .
(b) Suppose the estimator $\hat{\lambda}_2=1 / 9 \min \left(X_1, \ldots, X_9\right)$ is used to estimate the arrival rate. Find the probability that the estimator differs from the true arrival rate by more than 1.
(c) Generate 100 random samples of size 9. Compare the probabilities obtained in parts a and b to the observed relative frequencies.

Key Concepts

Monte Carlo Simulation Techniques

Monte Carlo simulation involves generating random samples repeatedly to approximate the behavior of a statistic or an estimator. This approach is especially useful when theoretical calculations become complex or when one wants to verify analytical results. By simulating numerous samples and computing the relative frequency of events, one can gain empirical insights into the accuracy and distributional properties of proposed estimators, providing a practical check against theoretical probability estimates.

Order Statistics in Exponential Distributions

Order statistics refer to the sorted values from a random sample, and the minimum of a set of independent exponential random variables is itself exponentially distributed with a rate that is the sum of the individual rates. This property is essential when constructing estimators based on the minimum value of the sample, as it allows the derivation of the distribution of the estimator and subsequent probability calculations regarding its deviation from the true parameter.

Estimator Accuracy: Probability of Large Deviations

Evaluating estimators often involves measuring the probability that the estimator deviates from the true parameter by more than a specified threshold. This concept is critical in statistical estimation and hypothesis testing, where such probabilities help assess estimator performance, reliability, and robustness. In problems like the one given, it translates to calculating the tail probabilities of the relevant distributions derived from the sample mean or the minimum statistic.

Exponential Distribution

The exponential distribution is a continuous probability distribution often used to model the times between events in a Poisson process. It is characterized by a constant hazard rate and has the memoryless property, meaning that the probability of an event occurring in the future is independent of the past. Its probability density function is defined by a rate parameter, and key properties like the mean and variance are directly related to this parameter.

Sample Mean as an Estimator and the Gamma Distribution

When dealing with samples from an exponential distribution, the sample mean is a natural estimator for the underlying parameter, and its distribution is related to the Gamma distribution. In many problems, including this one, the inverse of the sample mean is used to estimate the rate parameter. The Gamma distribution arises because the sum of independent exponentially distributed random variables follows a Gamma distribution, which provides the foundation for calculating probabilities and confidence intervals for such estimators.

Transcript

00:01 This question is given with an example which follows by zone distribution and we are asked to determine maximum likelihood estimate for theta and show that it is an unbiased estimate.

00:14 So the probability mass function for x is the probability mass function for x is equal to x theta is equal to we have to copy the theta power theta power x divide of our x equal to 1 3 3 the natural the natural log the natural law applies the function and the likelihood function is likely function is likely function is d a 4 n of theta is equal to ln of n theta x1 x2 up to x2 up to xm which is equal to n and of b power and theta theta power x1 plus x2 plus x3 up to xm divide the 5 x1 victorial x2 factorial x3 up to xm up to x so so this is equal to minus n theta plus n x bar l and of theta minus n of x1 victoria x2 factorial x3 factorial up to x n -2 here x1 plus x2 plus x3 plus x -3 plus x -n is equal to n x differentiation gives us differentiating the above statement or the above equation will give us differentiation gives lm ha is equal to theta is a quarter minus m theta minus x bar divided by theta solving of theta a time of theta minus theta qta q qa minus x bar divided by theta kore is equal to 0 given theta is equal to since l i mat theta is equal to minus x sorry minus n x bar divided by theta is equal to sorry this then zero we see that theta cape magazine mize and of theta kep, therefore the maximum is theta care which is equal to x bar which is equal to sum of x i i from 1 to x i divided by so so the parameter theta is the expected value of the present and the medium but we see that expected value of theta cap you see that expected value of theta cap is equal to expected value of x -by is equal to expected value of x which is equal to theta so it shows that the mn e estimated for theta is unbiased so m -l -e estimator for theta is embraced.

11:02 Now coming to the next part of the question, b part of the question...

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