# Continuous Random Variables and Probability Distributions

Chapter Outline

4-1 Continuous Random Variables

4-2 Probability Distributions and Probability Density Functions

4-3 Cumulative Distribution Functions

4-4 Mean and Variance of a Continuous Random Variable

4-5 Continuous Uniform Distribution

4-7 Normal Approximation to the Binomial and Poisson Distributions

The kinetic theory of gases provides a link between statistics and physical phenomena. The physicist James Maxwell used some basic assumptions to determine the distribution of molecular velocity in a gas at equilibrium. As a result of molecular collisions, all directions of rebound are equally likely. From this concept, he assumed equal probabilities for velocities in all the *x*, *y*, and *z* directions and independence of these components of velocity. This alone is sufficient to show that the probability distribution of the velocity in a particular direction *x* is the continuous probability distribution known as the normal distribution. This fundamental probability distribution can be derived from other directions (such as the central limit theorem to be discussed ...

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