Introduction
The Bernoulli distribution is a discrete probability distribution renowned for its simplicity and applicability to scenarios with two possible outcomes.
Step 1 — Understanding Bernoulli Distribution
The Bernoulli distribution thrives on two potential outcomes, 1 for success with a probability of 'p’ and 0 for failure with a probability of 'q’ where q = 1 - p. This distribution assumes a single experiment generating either 0 or 1 exemplifying the essence of a Bernoulli random variable.
Step 2 — Probability Characteristics
In this context, the probability of success (p) stands for the chance of achieving the desired outcome while the probability of failure q or 1 - p signifies the likelihood of not achieving the outcome.
Step 3 — Mathematical Expression
Mathematically, the probability function of a Bernoulli random variable is expressed as:
Step 4 — Bernoulli Trial and Process
A single instance of an experiment with two feasible results constitutes a Bernoulli trial. Conversely, a sequence of such outcomes constitutes a Bernoulli process.
Step 5 — Expected Value
For a Bernoulli random variable, the expected value (E(X)) equals the probability of success (p). In a scenario where p = 0.15 and q = 0.85, the expected value equals the mean of the distribution.
Step 6 — Python Implementation
To witness the Bernoulli distribution in action, let’s implement it using Python. The Python program uses the SciPy library to simulate and visualize both fair and unfair coins.
Step 7 — Interpreting the Results
By running the experiment 1,000 times (as indicated in line 5), we validate the behavior of the Bernoulli distribution. A probability value of p = 0.5 in line 7 signifies a fair coin with balanced chances of heads and tails. Conversely, p = 0.2 in line 16 denotes an unfair coin, skewing the probabilities. The generated plots in the output visualize the distribution outcomes, corroborating our analysis.
Conclusion
The Bernoulli distribution, a cornerstone of probability theory, thrives on its simplicity and real-world relevance. Through our comprehensive guide and illustrative Python code, we have navigated the intricacies of this distribution with precision. Armed with this understanding you can confidently apply Bernoulli principles to various scenarios, enabling data driven decision making.
