Standard Probability Distributions

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Standard Probability Distributions

Probability distributions are the foundation of statistical analysis and machine learning. They help us model real-world phenomena, make predictions, and understand uncertainty. Among the most fundamental distributions are the Bernoulli distribution (for discrete random variables) and its continuous counterparts. In this blog, we’ll dive into the Bernoulli distribution, explore its properties, and introduce other standard discrete and continuous distributions.

1. Bernoulli Distribution

The Bernoulli distribution is the simplest discrete probability distribution, modeling a random experiment with exactly two outcomes: success (1) and failure (0).

Definition:

A random variable ( X ) follows a Bernoulli distribution with parameter ( p ) (the probability of success) if:
[
P(X = 1) = p \quad \text{and} \quad P(X = 0) = 1 – p
]
Here, ( p ) is a number between 0 and 1.

Probability Mass Function (PMF):

[
P(X = x) = p^x (1 – p)^{1 – x} \quad \text{for } x \in {0, 1}
]

Properties:

Mean (Expected Value): ( E[X] = p )
Variance: ( \text{Var}(X) = p(1 – p) )
Applications: Modeling binary outcomes like coin flips, yes/no decisions, or success/failure experiments.

Example:

Consider flipping a fair coin. Let ( X = 1 ) if the outcome is heads and ( X = 0 ) if it’s tails. Then:
[
P(X = 1) = 0.5 \quad \text{and} \quad P(X = 0) = 0.5
]

2. Other Standard Discrete Distributions

a. Binomial Distribution

The binomial distribution models the number of successes in ( n ) independent Bernoulli trials.

PMF:
[
P(X = k) = \binom{n}{k} p^k (1 – p)^{n – k} \quad \text{for } k = 0, 1, \dots, n
]
Mean: ( E[X] = np )
Variance: ( \text{Var}(X) = np(1 – p) )
Example: Number of heads in 10 coin flips.

b. Poisson Distribution

The Poisson distribution models the number of events occurring in a fixed interval of time or space.

PMF:
[
P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!} \quad \text{for } k = 0, 1, 2, \dots
]
Mean: ( E[X] = \lambda )
Variance: ( \text{Var}(X) = \lambda )
Example: Number of emails received in an hour.

c. Geometric Distribution

The geometric distribution models the number of trials until the first success in a sequence of Bernoulli trials.

PMF:
[
P(X = k) = (1 – p)^{k – 1} p \quad \text{for } k = 1, 2, 3, \dots
]
Mean: ( E[X] = \frac{1}{p} )
Variance: ( \text{Var}(X) = \frac{1 – p}{p^2} )
Example: Number of coin flips until the first heads.

3. Standard Continuous Distributions

a. Uniform Distribution

The uniform distribution models equal likelihood over a continuous interval.

PDF:
[
f(x) = \frac{1}{b – a} \quad \text{for } x \in [a, b]
]
Mean: ( E[X] = \frac{a + b}{2} )
Variance: ( \text{Var}(X) = \frac{(b – a)^2}{12} )
Example: Selecting a random number between 0 and 1.

b. Normal (Gaussian) Distribution

The normal distribution is one of the most important distributions, characterized by its bell-shaped curve.

PDF:
[
f(x) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{(x – \mu)^2}{2\sigma^2}}
]
Mean: ( E[X] = \mu )
Variance: ( \text{Var}(X) = \sigma^2 )
Example: Heights of people, measurement errors.

c. Exponential Distribution

The exponential distribution models the time between events in a Poisson process.

PDF:
[
f(x) = \lambda e^{-\lambda x} \quad \text{for } x \geq 0
]
Mean: ( E[X] = \frac{1}{\lambda} )
Variance: ( \text{Var}(X) = \frac{1}{\lambda^2} )
Example: Time between arrivals at a service center.

4. Applications of Standard Distributions

Bernoulli and Binomial:

Modeling binary outcomes (e.g., success/failure, yes/no).
Example: A/B testing, quality control.

Poisson:

Counting rare events (e.g., number of accidents, website visits).
Example: Traffic flow analysis, call center modeling.

Geometric:

Modeling waiting times until the first success.
Example: Reliability testing, game design.

Uniform:

Modeling equal likelihood over an interval.
Example: Random number generation, simulations.

Normal:

Modeling natural phenomena and measurement errors.
Example: Statistical inference, machine learning.

Exponential:

Modeling time-to-event data.
Example: Queueing theory, survival analysis.

5. Key Takeaways

The Bernoulli distribution is the foundation for modeling binary outcomes.
Other discrete distributions (e.g., binomial, Poisson, geometric) extend the Bernoulli to more complex scenarios.
Continuous distributions (e.g., uniform, normal, exponential) model a wide range of real-world phenomena.
Understanding these distributions is essential for data analysis, modeling, and decision-making.

Conclusion

Standard probability distributions like the Bernoulli, binomial, Poisson, uniform, normal, and exponential are indispensable tools for understanding and modeling randomness. Whether you’re analyzing data, building predictive models, or designing experiments, these distributions provide the mathematical framework to tackle real-world problems. By mastering them, you’ll be well-equipped to navigate the fascinating world of probability and statistics.

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