5. Use Excel function int(rand) 9)1 for generating uniformly distributed random

Question

5. Use Excel function int(rand) 9)1 for generating uniformly distributed random number from 1 to 9 with probability of 1,1 = 1, 2, 10, P(X i) (i)-1 /9. Make three tables size 50× 8, 250× 8, 1000 × 8 with uniformly distributed random numbers from 1 to 9. Find population probability distribution functions, means, variances and standard deviations and compare its with sample frequency distribution functions, averages, sample estimation of variances and standard deviations for each table. Draw the graphs. Estimate mean for the random variables in the obtained samples using 90% confidence intervals. Recalculate the tables 10 times and determine how many times the population mean does not belong to the obtained confident intervals. (5 points)

Explanation / Answer

For Gamma DIstribution with parameter k, the statistics are as follows:-

Mean = k

Variance = k

The probability distribution function of a Gamma DIstribution is given by

A random variable X has the standard gamma distribution with shape parameter k(0,)k(0,) if it has the probability density function f given by f(x)=1(k)xk1ex,0<x<

Check the link: http://www.math.uah.edu/stat/special/Gamma.html

Since the gamma distribution is largely a sum of 4 gamma distributions with parameter = 3, teh mean of the distribution should be 3X4 = 12.

For the simulation with 3 tables, we get the sample mean as 12.1288, which is pretty close to the theoritical mean.

The variance of standard Gamma distribution is k. In this case, it is sum of 4 distributions with variance 3. hence, the variance of teh distribution should be 12.However, teh variance from silulation is 36.

8.208274

Theta, 3 S No Col1 Col2 Col3 Col4 Col5 1 0.569417 4.575615 2.477288 1.37154 8.993861 2 0.874576 2.824774 0.125134 1.942747 5.767232 3 6.335615 4.508325 0.145218 0.601923 11.59108 4 2.30044 2.32731 0.958526 0.903516 6.489792 5 0.468077 12.45544 0.594905 0.260968 13.77939 6 2.325159 3.753836 12.00342 2.167333 20.24975 7 7.586946 11.45265 0.112486 4.641224 23.79331 8 2.091478 4.797574 0.571745 1.317828 8.778626 9 0.714812 3.522828 2.625886 0.642647 7.506174 10 2.245003 1.291724 2.161436 0.614238 6.312401 11 0.659594 3.064283 2.864353 1.620044

8.208274

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