Central limit theorem

If a random variable Y is the sum of n independent random variables that satisfy the following general conditions, for sufficiently large n, the variable Y is approximately normally distributed.

General conditions

1. The individual normal random variable, X_i , contribute a negligible amount to the variance of the sum of all the n random variables
2. It is also unlikely that a single random variable contributing significantly to the sum

If X₁ , X₂ , X₃ , …., X_n are the set of n independent random variables with i^th random variable having mean μ_i and variance σ²_i , the resulting variable from the summation of all the n random variables is approximately normally distributed with mean and standard deviation . The following standardized variable,

is an approximate standard normal distribution as n approaches infinity.

A linear transformation, Y = a + bX, of a random variable X with mean μ and variance σ² results in a normal variable Y with mean a+bμ and the variance b²σ² .

The Normal distribution

The most important distribution of a continuous random variable is the normal distribution. This has got several applications in Civil engineering. One of the illustrative applications in traffic engineering is with the distribution of the speed data. This distribution and its properties are very useful in dealing with the samples. It can also be used for approximating the other types of the probability distributions.

A continuous random variable X is said to follow the normal distribution if its pdf is

The cumulative distribution function for the normal variable is given below;