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Exponential Distribution - Stochastic Hydrology - Assignment, Exercises of Mathematical Statistics

Avinashilingam UniversityMathematical Statistics

The main points discuss in the assignment are: Exponential Distribution, Estimate Parameter, Maximum Likelihood Estimators, Parameters of Distribution, Method of Moments, Peak Flow, Standard Deviation, Chebyshev’s Inequality, Correlation Coefficient

Typology: Exercises

2012/2013

Uploaded on 04/20/2013

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Assignment – Module 3
1. A data set is assumed to follow exponential distribution, with
( )
1
x
fx e
α
α
−
=
x > 0
In the data set, 60% of the values are less than 2.5, estimate the parameter α.
2. Obtain the maximum likelihood estimators of the parameters a and b for the pdf.
( )
xa
b
e
fx b
−

−

=
a < x < ∞ , -∞ < a < ∞ , b > 0
3. Obtain the parameters of the following distribution using method of moments
f(x) = (a+1) x a 0 <x < 1
4. Peak flows at a site have a mean of 675 units and a standard deviation of 220 units.
Obtain the maximum probability that the peak flow in a year will deviate more than 450
units from the mean using Chebyshev’s inequality.
5. Obtain the correlation coefficient for the yearly rainfall and the yearly runoff of a
catchment, 20 years data for which is given below.
Rainfall (cm) 377 363 458 365 430 365 366 317 311 392
Runoff (cm) 365 357 416 358 399 358 359 328 324 375
Rainfall (cm) 353 439 410 423 436 601 336 464 490 402
Runoff (cm) 351 404 386 394 402 506 340 420 436 381
6. Consider the data in the previous problem; obtain the regression equation between
rainfall and runoff.
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Assignment – Module 3

  1. A data set is assumed to follow exponential distribution, with

f ( x ) = α^1 e − x^ α x > 0

In the data set, 60% of the values are less than 2.5, estimate the parameter α.

  1. Obtain the maximum likelihood estimators of the parameters a and b for the pdf.

x (^) ba f x e b

−^ − = a < x < ∞ , -∞ < a < ∞ , b > 0

  1. Obtain the parameters of the following distribution using method of moments f(x) = (a+1) x a^ 0 <x < 1
  2. Peak flows at a site have a mean of 675 units and a standard deviation of 220 units. Obtain the maximum probability that the peak flow in a year will deviate more than 450 units from the mean using Chebyshev’s inequality.
  3. Obtain the correlation coefficient for the yearly rainfall and the yearly runoff of a catchment, 20 years data for which is given below.

Rainfall (cm) 377 363 458 365 430 365 366 317 311 392 Runoff (cm) 365 357 416 358 399 358 359 328 324 375 Rainfall (cm) 353 439 410 423 436 601 336 464 490 402 Runoff (cm) 351 404 386 394 402 506 340 420 436 381

  1. Consider the data in the previous problem; obtain the regression equation between rainfall and runoff.

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  1. Observations for the past 10 years of groundwater discharge and estimated recharge (in consistent units) of an artesian aquifer are given below Discharge 12.2 10.4 10.6 12.6 14.2 13.0 14.0 12.0 10.4 11. Recharge 12.0 9.8 11.0 13.2 14.6 14.0 14.0 12.4 10.4 11. Assuming a linear relationship between discharge and recharge, with discharge as the dependent variable and recharge as independent variable, what will be the discharge when the recharge is 13 units?
  2. Generate 50 observations from an exponential distribution with λ = 0.7, Compare the simulated mean with observed mean.
  3. Generate independent samples of 10, 20, 50 and 100 observations from normal distribution with mean 5 and standard deviation 2. Compare these values of mean and standard deviation with the computed ones.
  4. Generate independent samples of 10, 20, 30, 40, 50 and100 observations from gamma distribution with η = 2 and λ = 2.
  5. Assuming that the peak flows (in cumecs) given below follow a (a) Normal Distribution, (b) log-normal distribution, and (c) Exponential Distribution, generate 50 values of the peak flows, and compare the appropriate statistics of the generated flows for each of the distributions used with those of the observed flows 14,000 17,700 17,500 15,500 20,500 18,100 15,800 14, 16,300 14,900 17,600 17,000 17,300 18,300 19,100 17, 19,400 22,900 16,200 14,

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