This project requires writing computer code to generate random variables of vari

Question

This project requires writing computer code to generate random variables of various kinds and measure their probability distributions. 1. Generate a sequence of each of the following types of random variables, each sequence should be at least 10,000 points long.

(a) A binomial random variable. Let the number of Bernoulli trials ben 12. Recall that the binomial random variable is defined as the number of 1's in n trials for a Bernoulli (binary) random variable. Let the parameter p in the Bernoulli trials be p 0.5109.

(b) A Poisson random variable as a limiting case of the binomial random variable with p 0.0125 or less and n 80 or more while maintaining a np 1

(c) A type 1 geometric random variable with parameter p 0.09

(d) A (continuous uniform random variable in the range [-2, 5]

(e) A Gaussian random variable with mean u 1.3172 and variance 1.9236. (f An exponential random variable with parameter 1.37

This project requires writing computer code to generate random variables of various kinds and measure their probability distributions. 1. Generate a sequence of each of the following types of random variables, each sequence should be at least 10,000 points long. (a) A binomial random variable. Let the number of Bernoulli trials ben 12. Recall that the binomial random variable is defined as the number of 1's in n trials for a Bernoulli (binary) random variable. Let the parameter p in the Bernoulli trials be p 0.5109. (b) A Poisson random variable as a limiting case of the binomial random variable with p 0.0125 or less and n 80 or more while maintaining a np 1 (c) A type 1 geometric random variable with parameter p 0.09 (d) A (continuous uniform random variable in the range [-2, 5] (e) A Gaussian random variable with mean u 1.3172 and variance 1.9236. (f An exponential random variable with parameter 1.37 2. Estimate the CDF and PDF or PMF as appropriate and plot these functions next to their theoretical counterparts. Compare and comment on the results obtained

Explanation / Answer

This project requires writing computer code to generate random variables of various kinds and measure their probability distributions. 1. Generate a sequence of each of the following types of random variables, each sequence should be at least 10,000 points long.

We can find sample using MINITAB.

(a) A binomial random variable. Let the number of Bernoulli trials ben 12. Recall that the binomial random variable is defined as the number of 1's in n trials for a Bernoulli (binary) random variable. Let the parameter p in the Bernoulli trials be p 0.5109.

Here n = 12 and p = 0.5109

steps :

Calc --> Random data --> Binomial --> Number of rows of data to generate : 10000 --> Store in columns(s) : C1

--> Number of trials : 12 --> Probability of success : 0.5109 --> ok

This will generate the size of 10,000 from Binomial distribution.

(b) A Poisson random variable as a limiting case of the binomial random variable with p 0.0125 or less and n 80 or more while maintaining a np 1

The Poisson(µ) distribution is the limit of the binomial(n, p) distribution with µ = np as n ? ?.

n = 80, p = 0.0125

Here µ = 80*0.0125 = 1

Calc --> Random data --> Poisson --> Number of rows of data to generate : 10000 --> Store in columns(s) : C2

--> mean : 1 --> ok

This will create sample of 10,000 from Poisson distribution.

(d) A (continuous uniform random variable in the range [-2, 5]

Calc --> Random data --> Uniform --> Number of rows of data to generate : 10000 --> Store in columns(s) : C3

-->Lower endpoint : -2 --> Upper endpoint : 5 --> ok

This will create random sample of size 10,000 from Uniform distribution.

(e) A Gaussian random variable with mean u 1.3172 and variance 1.9236.

Variance = 1.9236

sd = sqrt(1.9236) = 1.387

Calc --> Random data --> Normal --> Number of rows of data to generate : 10000 --> Store in columns(s) : C4

-->Mean : 1.3172 --> standard deviation : 1.387 --> ok

This will generate random sample of size 10000 from Normal distribution.

f. An exponential random variable with parameter 1.37

Calc --> Random data --> exponential --> Number of rows of data to generate : 10000 --> Store in columns(s) : C5 -->Scale : 1.37 --> Threshold : 0 --> ok

This will generate random sample of size 10,000 from Exponential distribution.

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