Continuous random variables

Continuous Random Variables

  • A continuous random variable can take any value within a range or interval.
  • All values in the specified range are possible outcomes, unlike discrete random variables where only specific values can occur.

Probability Density Function (PDF)

  • The probability of a continuous random variable taking an exact value is zero. This is because there are infinite possible outcomes within any range.
  • Instead of a Probability Mass Function as in discrete random variables, continuous random variables have a Probability Density Function (PDF).
  • The total area under the curve of a PDF sums to 1, representing the total probability of all outcomes.

Cumulative Distribution Function (CDF)

  • The cumulative distribution function (CDF) for continuous random variables gives the probability that the variable takes a value less than or equal to a certain value.
  • The CDF is the integral of the PDF from minus infinity to up to that certain value.

Uniform Distribution

  • A Uniform Distribution is defined over a continuous interval where all outcomes are equally likely.
  • A Uniform Distribution is defined by two parameters, ‘a’ and ‘b’, which are the smallest and largest possible outcomes respectively.

Parameters of the Uniform Distribution

  • The expected value (mean) of a uniform distribution is (a + b)/2.
  • This distribution has variance (b - a)² / 12, and the square root of the variance gives the standard deviation.

Normal Distribution

  • The normal distribution is the most common type of distribution assumed in technical stock market analysis and in other types of statistical analyses.
  • The normal distribution curve is bell-shaped and symmetrical around its mean µ.
  • It is defined by its mean µ (mu) and standard deviation 𝜎 (sigma). Together, they provide the complete characteristics of a normal distribution.

Properties of Normal Distribution

  • About 68% of values drawn from a normal distribution are within one standard deviation 𝜎 away from the mean; about 95% are within two standard deviations; and about 99.7% lie within three standard deviations.

Standard Normal Distribution

  • A standard normal distribution is a special case of the normal distribution where the mean µ is 0 and the standard deviation 𝜎 is 1.
  • Any normal distribution can be transformed to a standard normal distribution and vice versa using the formula Z = (X - µ) / 𝜎, where Z is a value from the standard normal distribution, X is a value from the original normal distribution, and 𝜎 is the standard deviation of the original normal distribution.

Application of Continuous Random Variables

  • Continuous random variables are used to model phenomena where data can take infinitely many values. Common examples include measurements such as height, weight, temperature, and time.