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. 123.00 330.33 108.91 145.11 204.91 114.79 182.34 127.96 60.84 164.88 89.59 109.11 299.87 111.10 200.19 302.74 185.36 117.64 210.07 311.13 386.22 150.58 125.86 94.33 262.09 100.85 203.24 280.55 247.11 66.24 285.37 An appropriate probability plot supports the use of the lognormal distribution as a reasonable model for stream flow. (a) Estimate the parameters of the distribution. [Hint: Remember that X has a lognormal distribution with parameters μ and σ^2 if ln(X) is normally distributed with mean μ and variance σ^2.] (Round your estimate for the mean to three decimal places, and round your estimate for the variance to four decimal places.) (b) Use the estimates of part (a) to calculate an estimate of the expected value of stream flow. [Hint: What is E(X)?] (Round your answer to two decimal places.)
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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 $E(X) ? ]$
The Basics of Statistical Inference
Point Estimation
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. 169.14 109.11 204.91 285.37 114.79 117.64 127.96 66.24 262.09 125.86 210.07 113.10 87.24 302.74 150.58 280.55 94.33 77.44 185.36 120.66 145.11 200.19 100.85 311.13 538.02 108.91 299.87 330.33 247.11 203.24 89.59 An appropriate probability plot supports the use of the lognormal distribution as a reasonable model for stream flow. (a) Estimate the parameters of the distribution. [Hint: Remember that X has a lognormal distribution with parameters μ and σ² if ln(X) is normally distributed with mean μ and variance σ².] (Round your estimate for the mean to three decimal places, and round your estimate for the variance to four decimal places.) (b) Use the estimates of part (a) to calculate an estimate of the expected value of stream flow. [Hint: What is E(X)?] (Round your answer to two decimal places.)
Madhur L.
An article in Journal of Hydrology ["Use of a Lognormal Distribution Model for Estimating Soil Water Retention Curves from Particle-Size Distribution Data" $(2006,$ Vol. $323(1),$ pp. $325-334)]$ considered a lognormal distribution model to estimate water retention curves for a range of soil textures. The particle-size distribution (in centimeters) was modeled as a lognormal random variable $X$ with $\theta=-3.8$ and $\omega=0.7 .$ Determine the following: (a) $P(X<0.02)$ (b) Value for $x$ such that $P(X \leq x)=0.95$ (c) Mean and variance of $X$
Continuous Random Variables and Probability Distributions
Lognormal Distribution
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