Continuous Distributions

Continuous Distributions

1. Continuous random variable

   • We are no longer interested in the probability of a single variable
   • we are now interested in the probability of variables located within a specific region or interval
   • uniform distribution
     • A continuous random variable has a uniform distribution if its values spread evenly over the range of possibilities
     • A uniform continuous distribution must have two parameters, the minimum possible value (a) and the maximum possible value (b)

2. density curve

   • For continuous random variable, we use density curve to depict the probability distribution
   • A density curve is a graph, which must satisfy two properties
     • The total area under the curve must equal one
     • Every point on the curve must have a vertical height that is zero or greater
       • The curve cannot fall below the x-axis

3. Normal Distribution

   • If a continuous random variable has a distribution with a graph that is symmetric and bell-shaped, we call it that it is a normal distribution
   • Characteristics:
     • The distribution is symmetric.
     • Normal probability distributions is defined by its mean µ and its standard deviation σ
     • The highest point on the normal curve is at the mean, which is also the median and mode.
     • The standard deviation determines the width of the curve.
     • Probabilities for the normal random variable are given by areas under the curve. The total area under the curve is 1
   • Standard Normal Distribution
     • Characteristics:
       • Mean equal to zero
       • Standard deviation equal to one
     • The letter z is used to designate the standard normal random variable.
   • Non-standard Normal Distribution
     • Mean of variables:
     • Variance of variables:
     • Mean of z for all variables:
     • Variance of z for all variables:
     • A non-standard normal distribution can be converted to a standard normal distribution of its z score values