This page explains the behavior of mean and variance over time for both **Bernoulli processes** and **random walks**. It discusses the differences between absolute and relative frequency trajectories, along with the impact of different distributions.
1. Mean and Variance in Bernoulli Processes
Absolute Frequency:
- Mean: In a Bernoulli process, the mean represents the cumulative number of successes over time. It increases steadily as each trial potentially adds successes, growing proportionally to the success probability \( p \).
- Variance: Variance also increases over time, widening as the number of trials increases. It grows at a slower rate than the mean due to the factor \( p(1 - p) \), which accounts for the probability of both success and failure.
Relative Frequency:
- Mean: The mean for relative frequency tends to stabilize as the number of trials increases. The trajectory of the mean starts off volatile but converges towards \( p \) as the number of trials grows.
- Variance: The variance decreases over time, as relative frequency becomes more stable and centered around \( p \), with fluctuations diminishing as the number of trials increases.
2. Mean and Variance in Random Walks
Absolute Frequency:
- Mean: In random walks, the mean tracks the net movement. Over time, the expected mean stays close to zero (if there is no bias), but individual realizations can diverge widely.
- Variance: Variance grows quadratically with time since the walk spreads out further from the origin with each additional step.
Relative Frequency:
- Mean: Like the Bernoulli case, the relative frequency in random walks stabilizes over time, reflecting the expected probability of a step in either direction.
- Variance: Variance in relative frequency diminishes over time as fluctuations average out. In a large number of trials, the distribution becomes centered around the expected proportion.
3. Comparing Distributions
Absolute Number of Successes vs Relative Frequencies:
- Absolute Number of Successes: The absolute distribution of successes widens with time, becoming more spread out as successes accumulate. The mean increases steadily, while the variance grows as more trials occur.
- Relative Frequencies: The relative frequency distribution, on the other hand, becomes tighter over time, with fluctuations shrinking as trials accumulate. The mean of the relative frequency converges toward the true probability \( p \) as the number of trials increases, while variance decreases.