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1. Concepts & Definitions
1.1. Continous random distribution of probability
1.2. Normal distribution of probability
1.3. Standard normal distribution of probability
1.4. Inverse standard normal distribution
1.6. Inverse Student's T distribution
2. Problem & Solution
2.1. Weight, dimension, and value per HS6
2.2. How to fit a distribution
2.3. Employing standard deviation
2.4. Total time spent in a system
Finding Student's T distribution
Finding Student's T distribution
It is denoted as X ∼ t(k). And is read as X is a continuous random variable that follows Student’s T distribution with parameter k. Where k is the degrees of freedom. If the sample size is n, then k = n-1. Student’s T distribution is a small sample size approximation of a normal distribution. As the degrees of freedom increase, the t distribution tends to become the Standard Normal distribution.
The Student’s T distribution PMF formula follows the next equation.
The Student’s T distribution CDF formula follows the next equation.
Instead of employing the previous equations, the values from tables or from computational commands are employed in problems that need Student's T distribution.
Visually, the Student’s t distribution looks much like a normal distribution but generally has fatter tails. Fatter tails as you may remember allows for a higher dispersion of variables, as there is more uncertainty.
The second characteristic of the Student’s t-statistic is that there are degrees of freedom. Usually, for a sample of n, we have n-1 degrees of freedom. So, for a sample of 20 observations, the degrees of freedom are 19.
Computational Experiment Example
Computational Experiment Example
The following code shows how to compute Student's T probabilities employing PDF of Normal distribution using norm.ppf command.
from scipy.stats import t
import matplotlib.pyplot as plt
import numpy as np
#creating an array of values between
#-3 to 3 with a difference of 0.2
x = np.arange(-3, 3, 0.2)
deg_f = 10
y = t.pdf(x, df = deg_f)
plt.plot(x,y,'r-',x, y,'bo')
plt.show()
plt.bar(x, y)
plt.show()
The following code is particularly interesting to show how to compute the interval of probabilities employing the cumulative distribution function (CDF) of Student's T distribution using t.cdf command.
from scipy.stats import t
import matplotlib.pyplot as plt
import numpy as np
#creating an array of values between
#-3 to 3 with a difference of 0.2
x = np.arange(-3, 3, 0.2)
mean = 0
std = 1
deg_f = 10
y = t.cdf(x, df = deg_f, loc = mean, scale = std)
plt.grid()
plt.plot(x, y)
plt.show()
Applying code in the numerical example
Applying code in the numerical example
The following code shows the probability of the numerical example employing CDF of Normal distribution using norm.cdf command.
from scipy.stats import t
import matplotlib.pyplot as plt
import numpy as np
x = [-3, 3]
mean = 0
std = 1
deg_f = 10
y = t.cdf(x, df = deg_f, loc = mean, scale = std)
print(y)
[0.00667183 0.99332817]
pinterval = y[1] - y[0]
print('P(-3 <= X <= 3) = ',pinterval)
P(-3 <= X <= 3) = 0.9866563449774304
The previous complete code is available in the following link:
https://colab.research.google.com/drive/1lCaiy-2PJvrgPbCKoKyRoLTA_y5r_Ne6?usp=sharing
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