From Parselib Import Counts The Number Of Successes In N Ind ✓ Solved

From Parselib Import Counts The Number Of Successes In N Ind

Counts the number of successes in n independent trials when the probability of success each time is pr.

[X] is binomially-distributed with n trials, pr probability.

n = 30

pr = 0.32

rv = stats.binom(n, pr)

Specify the x-values of interest:

t = range(0, n+1)

New figure:

figure()

grid(True)

Draw the PMF at the specified x-values:

X_pmf = rv.pmf(t)

title('PMF of a Binomial distribution')

xlabel('Value, X')

ylabel('Probability, P(X)')

stem(t, X_pmf)

From parselib import *

Generate an RV-stub for a Chi-squared continuous random variable:

[X] is Chi2-distributed with [dof] degrees of freedom.

dof = 5

sigma = 2.7

rv = stats.chi2(dof, 0, sigma)

Specify the x-values of interest:

bins = 100

t = linspace(0, 3*dof, bins)

New figure:

figure()

grid(True)

Draw the PDF at the specified x-values:

X_pdf = rv.pdf(t)

title('PDF of a Chi2 distribution')

xlabel('Value, X')

ylabel('Probability, P(X)')

plot(t, X_pdf)

Paper For Above Instructions

The purpose of this paper is to explore the implementation of statistical distributions using the Parselib and Scipy libraries in Python. In particular, we will focus on two prominent distributions: the Binomial distribution and the Chi-squared distribution. Statistical distributions are fundamental in various fields including economics, biology, and engineering because they help to model and analyze random phenomena.

Binomial Distribution

The Binomial distribution models the number of successes in a fixed number of independent Bernoulli trials. It is characterized by two parameters: the number of trials \(n\) and the probability of success \(p\). In this instance, we shall investigate a Binomial distribution with \(n = 30\) trials and a probability of success \(p = 0.32\). The probability mass function (PMF) of a Binomial distribution provides the probability of obtaining exactly \(k\) successes in \(n\) trials.

The following code allows us to represent this distribution graphically. First, we set the number of trials and the probability of success:

n = 30


pr = 0.32

We can then generate the random variable using:

rv = stats.binom(n, pr)

To evaluate the PMF, we can specify a range of values of interest:

t = range(0, n + 1)


X_pmf = rv.pmf(t)

This code calculates the PMF for each value in the specified range. Once the probabilities are computed, we can visualize the PMF using the following commands:

figure()


grid(True)


stem(t, X_pmf)


title('PMF of a Binomial distribution')


xlabel('Value, X')


ylabel('Probability, P(X)')

This graphical representation provides insights into the likelihood of achieving different counts of successes across our trials.

Chi-Squared Distribution

The Chi-squared distribution is a special case of the gamma distribution and is widely used in hypothesis testing and in constructing confidence intervals for variance. This distribution is particularly relevant when analyzing the variances of the means from categorical data. It is characterized by degrees of freedom, denoted as \(dof\).

In our example, we will explore a Chi-squared distribution with \(dof = 5\) degrees of freedom and a scale parameter \( \sigma = 2.7\). To model our random variable, we launch the following commands:

rv = stats.chi2(dof, 0, sigma)

Next, we set the range for evaluating the probability density function (PDF). To achieve this, we can choose a number of bins:

bins = 100


t = linspace(0, 3*dof, bins)

Then, we compute the PDF over the specified range of values:

X_pdf = rv.pdf(t)

We can visualize this distribution using:

figure()


grid(True)


plot(t, X_pdf)


title('PDF of a Chi2 distribution')


xlabel('Value, X')


ylabel('Probability, P(X)')

Visualizing the Chi-squared distribution can provide insights regarding the typical behavior of the variance among the observed data as well as assessing how well our statistical model fits the observed data.

Conclusion

In summary, the implementation of the Binomial and Chi-squared distributions using the Parselib and Scipy libraries illustrates the effectiveness of Python in performing statistical analyses. By leveraging built-in functions such as PMF and PDF, researchers and practitioners can efficiently visualize and interpret the behavior of random variables. This process is crucial in understanding data distribution, hypothesis testing, and building statistical models.

References

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