Introduction:

Probability distributions play a pivotal role in understanding the likelihood of different outcomes in various scenarios. Let's delve into two essential discrete non-Gaussian distributions: Bernoulli and Binomial.

1. Bernoulli Distribution:

a. Definition:

• • The Bernoulli Distribution represents a discrete probability distribution of a random variable taking either of two possible outcomes, typically labeled as success (1) or failure (0).

b. PDF and CDF Equations:

• • PMF: ( \(P(X=x) = p^x (1-p)^{1-x}\) ), where (p) is the probability of success.

    • CDF: ( \(F(x) = \begin{cases} 0 & \text{for } x < 0 \ 1-p & \text{for } 0 \leq x < 1 \ 1 & \text{for } x \geq 1 \end{cases}\))

c. Parameters:

• • (p) is the probability of success.

d. Example Usage:

• • Modeling outcomes with binary results, such as coin flips or success-failure experiments.

2. Binomial Distribution:

a. Definition:

• • The Binomial Distribution describes the number of successes in a fixed number of independent Bernoulli trials with the same probability of success.

b. PMF and CDF Equations:

• • PMF: ( \(P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}\) ), where (n) is the number of trials and (k) is the number of successes.

    • CDF: Cumulative distribution function calculated using the PMF.

• c. Parameters:

  • (n) is the number of trials, (p) is the probability of success.

• d. Example Usage:

  • Analyzing scenarios where there are a fixed number of independent trials with the same probability of success like the number of successful coin flips in a series.

Conclusion:

Understanding Bernoulli and Binomial distributions equips us to model discrete, binary outcomes with varying levels of complexity. From single trials with Bernoulli to multiple independent trials with Binomial, these distributions are foundational tools in probability theory and statistical modeling.