Consider the accompanying observations on stream flow...

Consider the accompanying observations on stream flow (thousands 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 Resources Res.). 127.96 285.37 200.19 125.86 117.64 204.91 94.33 210.07 100.85 66.24 114.79 302.74 311.13 203.24 89.59 247.11 109.11 280.55 150.58 108.91 185.36 299.87 330.33 145.11 262.09 178.21 126.94 109.64 85.54 95.36 477.08 An appropriate probability plot supports the use of the lognormal distribution ( see Sect. 3.5$)$ 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 $\sigma$ if $\ln (X)$ is normally distributed with mean $\mu$ and standard deviation $\sigma . ]$ (b) Use the estimates of part (a) to calculate an estimate of the expected value of stream flow $\quad$ [Hint: What is the expression for $E(X) ? ]$

Consider the accompanying observations on stream flow (thousands 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 Resources Res.).
127.96
285.37
200.19
125.86
117.64
204.91
94.33
210.07
100.85
66.24
114.79
302.74
311.13
203.24
89.59
247.11
109.11
280.55
150.58
108.91
185.36
299.87
330.33
145.11
262.09
178.21
126.94
109.64
85.54
95.36
477.08
An appropriate probability plot supports the use of the lognormal distribution ( see Sect. 3.5$)$ 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 $\sigma$ if $\ln (X)$ is normally distributed with mean $\mu$ and
standard deviation $\sigma . ]$
(b) Use the estimates of part (a) to calculate an estimate of the expected value of stream flow
$\quad$ [Hint: What is the expression for $E(X) ? ]$

Key Concepts

Expected Value Calculation

The expected value of a lognormal distribution can be calculated using the formula E(X) = exp(? + ?²/2), where ? and ? are the mean and standard deviation of the logarithmically transformed variable. This relationship arises from the properties of the exponential function applied to a normally distributed variable.

Parameter Estimation

Parameter estimation involves using sample data to estimate the parameters of a probability distribution. For a lognormal distribution, one estimates the mean (?) and standard deviation (?) of the normal distribution associated with the logarithms of the original data by computing the sample mean and standard deviation of the log-transformed observations.

Probability Plot

A probability plot is a graphical technique for assessing whether or not a data set follows a given distribution. By plotting the transformed data against the theoretic quantiles of the normal distribution, one can visually inspect the linearity that would confirm the appropriateness of the lognormal model.

Lognormal Distribution

A lognormal distribution is one where the logarithm of the variable is normally distributed. This means that while the original variable is skewed and positive, its natural log exhibits the familiar bell-shaped curve of a normal distribution.

Transformation to Normality

Transformation to normality refers to the process of applying a function (typically the natural log) to data so that the transformed data follows a normal distribution. This is particularly useful when modeling data that is multiplicative or skewed, enabling the application of statistical techniques that assume normality.

Transcript

00:01 In problem 6, we are told to consider the observations on stream flow, which is given in thousands of acre feet, recorded at a station in colorado over 31 -year span.

00:16 And we're told that an appropriate probability plot supports the use of the log normal distribution as a reasonable model for stream flow.

00:27 So we have some data showing the stream flow.

00:31 Flows and we are going to be using this data to estimate the parameters of the distribution this time the log normal distribution so you need to remember that in a log normal distribution the parameters are a mu and lambda and or sigma sorry and the natural logarithm of x is normally distributed with the mean mu and standard deviation signal so in this case the first part of the question if you're going to estimate the parameters then we're going to be looking at the following parameters it's going to be we're going to be looking at mu heart and sigma heart so mu heart is going to be based on then the natural logarithm of all of these values so we have to introduce the natural logarithms for of these values.

01:35 So if for example for 127 the natural logarithm will be 4 .85.

01:49 For 210 .07 the natural logarithm will be 5 .34 and so on.

01:59 So to get the estimate of the of mu we will have to work out the value of x bar the sample mean for the natural logarithm.

02:10 So we need to add up all the values and divide that by the sample size.

02:17 And in this case, so x bar, so in this case when we sum up all the values of the natural logarithm of x, you will get 158 .15 divided by the sample size, which is 31...

Please give Ace some feedback