Generate A Stream Of I.i.d. Random Variables U = {Un | 1 ≤

Generate a stream of i.i.d. random variables U = {Un | 1 ≤ n ≤ N} which are uniformly distributed over [0, 1], for each N = 1000, 5000, 10000.

Assignment Instructions: Generate and analyze sequences of independent and identically distributed (i.i.d.) uniform random variables over the interval [0, 1], specifically for sample sizes N = 1000, 5000, and 10000. Use these sequences to create histograms of the probability density over ten equal intervals (0, 0.1], (0.1, 0.2], ..., (0.9, 1.0], and compare the empirical histogram levels with the theoretical uniform density. Then, compute the sample means and variances for each sequence and compare these to the theoretical mean and variance of a uniform distribution (0.5 and 1/12, respectively). Additionally, demonstrate how transforming uniform variables via Tn = -ln(Un)/λ results in exponential distributions with rate λ. Include this theoretical transformation and its properties in your analysis.

Paper For Above instruction

Introduction

The generation and analysis of random variables are fundamental in understanding stochastic processes and their applications, ranging from simulation modeling to statistical inference. The uniform distribution serves as one of the most basic and versatile distributions in probability theory, providing raw material for generating more complex distributions, such as the exponential. This paper details the process of generating i.i.d. uniform random variables for differing sample sizes, analyzing their empirical distributions through histograms, and verifying theoretical properties through statistical measures. Furthermore, the transformation from uniform to exponential variables illustrates the fundamental relationship between these distributions, essential in fields like reliability analysis, queuing theory, and network traffic modeling.

Generation of Uniform Random Variables and Histogram Analysis

To generate the uniform random variables, a pseudo-random number generator (PRNG) such as the Mersenne Twister can be employed, ensuring that the generated streams are statistically independent and uniformly distributed over [0, 1]. For each of the specified sample sizes (N=1000, 5000, 10000), sequences U = {Un} are generated, and their distributions are examined visually and statistically. The probability density function (PDF) of a uniform distribution is constant over its range, with a theoretical density of 1 within [0, 1]. Accordingly, the empirical PDFs are visualized via histograms segmented into ten equal intervals, allowing for comparison with the analytical constant density level.

Histograms are constructed by counting the number of samples falling into each interval, normalized by the total number of samples and the interval width. The levels of these histograms should approximate the constant density value of 1 as the sample size increases, illustrating the Law of Large Numbers. For N = 1000, some fluctuations are expected, while at N=10,000, the histogram should closely align with the theoretical density, demonstrating the convergence properties of empirical measures in large samples.

Analysis of Sample Means and Variances

The sample mean (μN) and variance (σ2N) are calculated for each dataset using the formulas:

μN = (1/N) Σ_{i=1}^{N} Un,
σ2N = (1/(N-1)) Σ_{i=1}^{N} (Un - μN)^2.

The theoretical mean of a uniform distribution over [0, 1] is 0.5, and its variance is 1/12 ≈ 0.0833. As the sample size increases, the empirical mean should converge towards 0.5, and the variance towards 1/12, demonstrating the consistency of the sample moments. The analysis involves comparing the computed sample moments for each N with the theoretical values, discussing deviations due to sampling variability, and emphasizing the effects of larger sample sizes on estimation accuracy.

Transformation to Exponential Distribution

A key property of uniform distributions is their role as raw inputs for transformation into other distributions. Specifically, if Un ~ Uniform(0, 1], then the random variable Tn = -ln(Un)/λ follows an exponential distribution with rate λ. This transformation is based on the relation between the cumulative distribution function (CDF) of the exponential and the uniform distribution:

F_T(t) = P(Tn ≤ t) = P(-ln(Un)/λ ≤ t) = P(Un ≥ e^{-λ t}) = 1 - e^{-λ t}.

Therefore, Tn reproduces the exponential distribution with the specified rate. Verifying this property involves generating uniform variables, transforming them, and analyzing the resulting distribution's histogram over intervals (i, i+1], for i=0,...,19. The empirical distribution should align with the theoretical exponential density, further illustrating the utility of the uniform distribution as a building block for more complex distributions.

Conclusion

The generation and analysis of uniform and exponential random variables exemplify core principles of probability and statistical inference. Empirical histograms affirm the uniform distribution's constant density and the exponential distribution's characteristic shape, with larger sample sizes providing closer approximations to theoretical distributions. The transformation from uniform to exponential demonstrates a vital functional connection utilized extensively in stochastic modeling. Overall, these procedures reinforce fundamental concepts of random variable generation, convergence, and the practical significance of theoretical distribution properties.

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