In this section, we will explore several other important probability distributions that are frequently used in statistical analysis. Understanding these distributions will help you model different types of data and perform more accurate statistical inferences.
Key Concepts
- Poisson Distribution
- Exponential Distribution
- Uniform Distribution
- Chi-Square Distribution
- t-Distribution
- F-Distribution
- Poisson Distribution
The Poisson distribution is used to model the number of events occurring within a fixed interval of time or space. It is particularly useful for modeling rare events.
Characteristics:
- Discrete distribution
- Describes the number of events in a fixed interval
- Events occur independently
Formula:
\[ P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!} \] Where:
- \( \lambda \) is the average number of events in the interval
- \( k \) is the number of events
- \( e \) is the base of the natural logarithm
Example:
import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import poisson
# Parameters
lambda_ = 3
# Generate Poisson distribution
x = np.arange(0, 15)
pmf = poisson.pmf(x, lambda_)
# Plot
plt.bar(x, pmf)
plt.title('Poisson Distribution (λ=3)')
plt.xlabel('Number of events')
plt.ylabel('Probability')
plt.show()Exercise:
Calculate the probability of observing exactly 4 events in an interval if the average number of events is 2.
Solution: \[ P(X = 4) = \frac{2^4 e^{-2}}{4!} = 0.090 \]
- Exponential Distribution
The exponential distribution is used to model the time between events in a Poisson process.
Characteristics:
- Continuous distribution
- Describes the time between events
- Memoryless property
Formula:
\[ f(x; \lambda) = \lambda e^{-\lambda x} \] Where:
- \( \lambda \) is the rate parameter
- \( x \) is the time between events
Example:
import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import expon
# Parameters
lambda_ = 1
# Generate Exponential distribution
x = np.linspace(0, 10, 1000)
pdf = expon.pdf(x, scale=1/lambda_)
# Plot
plt.plot(x, pdf)
plt.title('Exponential Distribution (λ=1)')
plt.xlabel('Time between events')
plt.ylabel('Probability Density')
plt.show()Exercise:
Calculate the probability that the time between events is less than 3 units if the rate parameter \( \lambda \) is 0.5.
Solution: \[ P(X < 3) = 1 - e^{-0.5 \times 3} = 0.776 \]
- Uniform Distribution
The uniform distribution is used to model a situation where all outcomes are equally likely within a certain range.
Characteristics:
- Continuous distribution
- All outcomes are equally likely within the range
Formula:
\[ f(x; a, b) = \frac{1}{b - a} \] Where:
- \( a \) and \( b \) are the lower and upper bounds
Example:
import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import uniform
# Parameters
a, b = 0, 10
# Generate Uniform distribution
x = np.linspace(a, b, 1000)
pdf = uniform.pdf(x, loc=a, scale=b-a)
# Plot
plt.plot(x, pdf)
plt.title('Uniform Distribution (a=0, b=10)')
plt.xlabel('Value')
plt.ylabel('Probability Density')
plt.show()Exercise:
Calculate the probability that a value is between 2 and 5 in a uniform distribution ranging from 0 to 10.
Solution: \[ P(2 \leq X \leq 5) = \frac{5 - 2}{10 - 0} = 0.3 \]
- Chi-Square Distribution
The chi-square distribution is used in hypothesis testing and constructing confidence intervals for variance.
Characteristics:
- Continuous distribution
- Sum of the squares of \( k \) independent standard normal variables
Formula:
\[ f(x; k) = \frac{1}{2^{k/2} \Gamma(k/2)} x^{(k/2)-1} e^{-x/2} \] Where:
- \( k \) is the degrees of freedom
- \( \Gamma \) is the gamma function
Example:
import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import chi2
# Parameters
k = 5
# Generate Chi-Square distribution
x = np.linspace(0, 20, 1000)
pdf = chi2.pdf(x, k)
# Plot
plt.plot(x, pdf)
plt.title('Chi-Square Distribution (k=5)')
plt.xlabel('Value')
plt.ylabel('Probability Density')
plt.show()Exercise:
Calculate the probability that a chi-square random variable with 3 degrees of freedom is less than 4.
Solution: \[ P(X < 4) = 0.608 \] (using chi-square cumulative distribution function)
- t-Distribution
The t-distribution is used in hypothesis testing for small sample sizes.
Characteristics:
- Continuous distribution
- Similar to the normal distribution but with heavier tails
Formula:
\[ f(x; \nu) = \frac{\Gamma((\nu+1)/2)}{\sqrt{\nu \pi} \Gamma(\nu/2)} \left(1 + \frac{x^2}{\nu}\right)^{-(\nu+1)/2} \] Where:
- \( \nu \) is the degrees of freedom
- \( \Gamma \) is the gamma function
Example:
import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import t
# Parameters
df = 10
# Generate t-distribution
x = np.linspace(-5, 5, 1000)
pdf = t.pdf(x, df)
# Plot
plt.plot(x, pdf)
plt.title('t-Distribution (df=10)')
plt.xlabel('Value')
plt.ylabel('Probability Density')
plt.show()Exercise:
Calculate the probability that a t-distributed random variable with 5 degrees of freedom is between -2 and 2.
Solution: \[ P(-2 < X < 2) = 0.857 \] (using t-distribution cumulative distribution function)
- F-Distribution
The F-distribution is used in analysis of variance (ANOVA) and regression analysis.
Characteristics:
- Continuous distribution
- Ratio of two chi-square distributions
Formula:
\[ f(x; d_1, d_2) = \frac{\sqrt{\left(\frac{d_1 x}{d_1 x + d_2}\right)^{d_1} \left(\frac{d_2}{d_1 x + d_2}\right)^{d_2}}}{x B(d_1/2, d_2/2)} \] Where:
- \( d_1 \) and \( d_2 \) are the degrees of freedom
- \( B \) is the beta function
Example:
import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import f
# Parameters
d1, d2 = 5, 2
# Generate F-distribution
x = np.linspace(0, 5, 1000)
pdf = f.pdf(x, d1, d2)
# Plot
plt.plot(x, pdf)
plt.title('F-Distribution (d1=5, d2=2)')
plt.xlabel('Value')
plt.ylabel('Probability Density')
plt.show()Exercise:
Calculate the probability that an F-distributed random variable with 3 and 4 degrees of freedom is less than 2.
Solution: \[ P(X < 2) = 0.684 \] (using F-distribution cumulative distribution function)
Conclusion
In this section, we covered several important probability distributions beyond the binomial and normal distributions. Each distribution has its own unique characteristics and applications. Understanding these distributions will enhance your ability to model and analyze different types of data effectively. In the next module, we will delve into statistical inference, where these distributions play a crucial role.