Sampling Mean (\(\bar{X}\))
The sampling mean is a measure of the "central tendency" of a dataset. It is calculated by summing all the observed values and dividing by the total number of observations. Mathematically:
\[ \bar{X} = \frac{1}{n} \sum_{i=1}^n X_i \]
- Intuitive Meaning: It represents the "average" value of a sample. If you randomly pick one value, it is likely to be close to the mean.
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Properties:
- \(E[\bar{X}] = \mu\): The sample mean is an unbiased estimator of the population mean.
- \(\text{Var}(\bar{X}) = \frac{\sigma^2}{n}\): The variability of the sample mean decreases as the sample size increases.
Sampling Variance (\(S^2\))
The sampling variance measures the spread of data, i.e., how far the observed values deviate from the sample mean. It is defined as:
\[ S^2 = \frac{1}{n-1} \sum_{i=1}^n (X_i - \bar{X})^2 \]
- Intuitive Meaning: It shows how much the data "varies" around the mean. If the data points are close to the mean, the variance is small; otherwise, it is large.
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Properties:
- \(E[S^2] = \sigma^2\): The sample variance is an unbiased estimate of the population variance.
- As the sample size increases, \(S^2\) gets closer to \(\sigma^2\).
The Law of Large Numbers (LLN)
The Law of Large Numbers states that as the sample size (\(n\)) increases, the sample mean (\(\bar{X}\)) approaches the true population mean (\(\mu\)).
Mathematical Statement
\[ \lim_{n \to \infty} P(|\bar{X}_n - \mu| < \epsilon) = 1, \quad \text{for any } \epsilon > 0. \]
This guarantees that, with enough observations, we can accurately estimate population parameters.
Illustrative Example
Suppose you roll a fair die. Each number (\(1, 2, 3, 4, 5, 6\)) has a probability of \(1/6\), and the theoretical mean is:
\[ \mu = \frac{1+2+3+4+5+6}{6} = 3.5 \]
Initially, the results of a few rolls might deviate significantly (e.g., \(1, 6, 2\)), but as the number of rolls increases, the average (\(\bar{X}\)) will converge to 3.5.
Applications in Cybersecurity
- Anomaly Detection: By collecting and analyzing large volumes of network traffic, baseline statistics can be established. Significant deviations from these baselines may indicate potential attacks.
- Encryption Validation: The LLN can test whether pseudo-random number generators produce uniformly distributed outputs, a critical property for secure cryptographic systems.
- Risk Modeling: Estimating the average time between security breaches helps in planning defensive strategies.
- Fraud Detection: Analyzing user behavior over time to identify unusual patterns indicative of fraudulent activities.
Exercise
Following the same scheme of HMWK 7 compute the distribution of the sampling variance ("corrected" or not). Determine the distribution of the variances of the samples, and its mean and variance. discussing the observed relationship with the mean and variance of the parent (theoretical) distribution.
The results include statistical summaries and visual representations to deepen understanding of these important statistical concepts.
Access the Exercise
You can access the full interactive exercise by clicking the link below: