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Chapter 6: Problem 6

Consider the accompanying observations on stream flow (1000s of acre-feet) recorded at a station in Colorado for the period April 1-August 31 over a 31-year span (from an article in the 1974 volume of Water 91影视 Research). $$ \begin{array}{rrrrr} 127.96 & 210.07 & 203.24 & 108.91 & 178.21 \\ 285.37 & 100.85 & 89.59 & 185.36 & 126.94 \\ 200.19 & 66.24 & 247.11 & 299.87 & 109.64 \\ 125.86 & 114.79 & 109.11 & 330.33 & 85.54 \\ 117.64 & 302.74 & 280.55 & 145.11 & 95.36 \\ 204.91 & 311.13 & 150.58 & 262.09 & 477.08 \\ 94.33 & & & & \end{array} $$ An appropriate probability plot supports the use of the lognormal distribution (see Section 4.5) as a reasonable model for stream flow. a. Estimate the parameters of the distribution. [Hint: Remember that \(X\) has a lognormal distribution with parameters \(\mu\) and \(\sigma^{2}\) if \(\ln (X)\) is normally distributed with mean \(\mu\) and variance \(\sigma^{2}\).] b. Use the estimates of part (a) to calculate an estimate of the expected value of stream flow.

Short Answer

Expert verified
1. Compute \(\ln(X)\) for each observation; 2. Calculate mean \(\bar{Y}\) and variance \(S_Y^2\) of the logarithms; 3. \(\mu = \bar{Y}, \sigma^2 = S_Y^2\); 4. \(E(X) = e^{\mu + \sigma^2/2}\).

Step by step solution

01

Calculate the Logarithms

To estimate the parameters of a lognormal distribution, first compute the natural logarithm of each stream flow observation. For each observation \(X\), calculate \(Y = \ln(X)\).
02

Calculate Sample Mean and Variance

Using the logarithmic values calculated in Step 1, find the sample mean \(\bar{Y}\) and the sample variance \(S_Y^2\). These statistics will be used to estimate the parameters \(\mu\) and \(\sigma^2\) of the normal distribution model for \(\ln(X)\).
03

Determine Parameters of Lognormal Distribution

The sample mean \(\bar{Y}\) is used as an estimate for \(\mu\), the mean of the normal distribution for \(\ln(X)\). The sample variance \(S_Y^2\) is used to estimate \(\sigma^2\), the variance of the normal distribution. Thus, \(\mu = \bar{Y}\) and \(\sigma^2 = S_Y^2\).
04

Estimate Expected Value of Stream Flow

For a lognormal distribution, the expected value \(E(X)\) is calculated using the formula \(E(X) = e^{\mu + \frac{\sigma^2}{2}}\). Substitute the estimates \(\mu\) and \(\sigma^2\) obtained from Step 3 into this formula to calculate the expected value of the stream flow.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Stream Flow Modeling
Stream flow modeling is the practice of using mathematical techniques to represent and predict the flow of water in rivers and streams. The goal is to understand the patterns in stream flows and forecast future flow conditions.
This process is crucial for water resource management, flood forecasting, and infrastructure planning. In this context, the stream flow data collected from a specific location over several years is used to create a probabilistic model that predicts future conditions. One commonly used distribution in stream flow modeling is the lognormal distribution. The lognormal distribution is chosen because of its ability to handle non-negative data skewed to the right, which is a common characteristic of stream flow data.
A probability plot can help decide on the best fit model by comparing how observed data aligns with a theoretical distribution. Understanding stream flow patterns helps in:
  • Managing water supplies
  • Planning agricultural activities
  • Designing hydraulic structures
Parameter Estimation
Parameter estimation involves determining the key parameters of a distribution that describes a dataset. In the lognormal distribution context, these parameters are \(\mu\) and \(\sigma^2\). These parameters represent the mean and variance of the natural logarithm of the data, rather than the data itself.
For estimation, the process generally involves:
  • Taking the natural logarithm of the observed data.
  • Computing the sample mean (\(\bar{Y}\)) and the sample variance (\(S_Y^2\)) of the logarithmic values.
These steps aim to use the observed data to estimate the underlying distribution parameters that best describe the data's behavior.
This method assumes that the logarithmic transformation turns the stream flow data into a normally distributed dataset, allowing the sample mean and variance calculations to estimate the parameters \((\mu, \sigma^2)\) required to model the data accurately.Parameter estimation plays a critical role in making accurate predictions in fields like hydrology and environmental science.
Expected Value Calculation
In probability and statistics, expected value is a foundational concept which provides the average or 'expected' outcome over numerous instances of a random event. For a lognormal distribution, the expected value takes into account the nature of the distribution, showing how far it deviates from just the mean of the natural logarithm of the values.
To calculate the expected value for a lognormal distribution, we use the formula:\[ E(X) = e^{\mu + \frac{\sigma^2}{2}} \]This formula incorporates both the mean and variance of the distribution of the natural logarithm of the data. By substituting the estimated mean (\(\mu\)) and variance (\(\sigma^2\)) into the formula, we can calculate the expected stream flow, offering insights into the average water discharge to be expected under similar conditions.The expected value calculation helps to provide:
  • A predictive average for planning resource management.
  • Baseline data for comparing actual versus expected conditions.
  • Foundational information for decision making in water resource projects.

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Most popular questions from this chapter

A sample of \(n\) captured Pandemonium jet fighters results in serial numbers \(x_{1}, x_{2}, x_{3}, \ldots, x_{n}\). The CIA knows that the aircraft were numbered consecutively at the factory starting with \(\alpha\) and ending with \(\beta\), so that the total number of planes manufactured is \(\beta-\alpha+1\) (e.g., if \(\alpha=17\) and \(\beta=29\), then \(29-\) \(17+1=13\) planes having serial numbers \(17,18,19, \ldots\), 28,29 were manufactured). However, the CIA does not know the values of \(\alpha\) or \(\beta\). A CIA statistician suggests using the estimator \(\max \left(X_{i}\right)-\min \left(X_{i}\right)+1\) to estimate the total number of planes manufactured. a. If \(n=5, x_{1}=237, x_{2}=375, x_{3}=202, x_{4}=525\), and \(x_{5}=418\), what is the corresponding estimate? b. Under what conditions on the sample will the value of the estimate be exactly equal to the true total number of planes? Will the estimate ever be larger than the true total? Do you think the estimator is unbiased for estimating \(\beta-\alpha+1\) ? Explain in one or two sentences.

A sample of 20 students who had recently taken elementary statistics yielded the following information on the brand of calculator owned \((\mathrm{T}=\) Texas Instruments, \(\mathrm{H}=\) Hewlett Packard, C = Casio, \(\mathrm{S}=\) Sharp): a. Estimate the true proportion of all such students who own a Texas Instruments calculator. b. Of the 10 students who owned a TI calculator, 4 had graphing calculators. Estimate the proportion of students who do not own a TI graphing calculator.

When the sample standard deviation \(S\) is based on a random sample from a normal population distribution, it can be shown that $$ E(S)=\sqrt{2 /(n-1)} \Gamma(n / 2) \sigma / \Gamma((n-1) / 2) $$ Use this to obtain an unbiased estimator for \(\sigma\) of the form \(c S\). What is \(c\) when \(n=20\) ?

The accompanying data on flexural strength (MPa) for concrete beams of a certain type was introduced in Example 1.2. $$ \begin{array}{rrrrrrr} 5.9 & 7.2 & 7.3 & 6.3 & 8.1 & 6.8 & 7.0 \\ 7.6 & 6.8 & 6.5 & 7.0 & 6.3 & 7.9 & 9.0 \\ 8.2 & 8.7 & 7.8 & 9.7 & 7.4 & 7.7 & 9.7 \\ 7.8 & 7.7 & 11.6 & 11.3 & 11.8 & 10.7 & \end{array} $$ a. Calculate a point estimate of the mean value of strength for the conceptual population of all beams manufactured in this fashion, and state which estimator you used. [Hint: \(\left.\Sigma x_{i}=219.8 .\right]\) b. Calculate a point estimate of the strength value that separates the weakest \(50 \%\) of all such beams from the strongest \(50 \%\), and state which estimator you used. c. Calculate and interpret a point estimate of the population standard deviation \(\sigma\). Which estimator did you use? d. Calculate a point estimate of the proportion of all such beams whose flexural strength exceeds \(10 \mathrm{MPa}\). [Hint: Think of an observation as a "success" if it exceeds 10.] e. Calculate a point estimate of the population coefficient of variation \(\sigma / \mu\), and state which estimator you used.

Two different computer systems are monitored for a total of \(n\) weeks. Let \(X_{i}\) denote the number of breakdowns of the first system during the \(i\) th week, and suppose the \(X_{i}\) 's are independent and drawn from a Poisson distribution with parameter \(\mu_{1}\). Similarly, let \(Y_{i}\) denote the number of breakdowns of the second system during the \(i\) th week, and assume independence with each \(Y_{i}\) Poisson with parameter \(\mu_{2}\). Derive the mle's of \(\mu_{1}, \mu_{2}\), and \(\mu_{1}-\mu_{2}\).
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