Statistics

Is variable that has infinite number of values. In other words, any value is possible for the variable.

These values are often associated with measurements on a continuous scale with no gaps or interruption, such as time based events (e.g. computer processing time).

For continuous random variables - In theory, the probability for any single value is equal to zero !! For example: Considering an uniform distribution, randomly select a value x from [0, 1], what is the probability P(X=x)?

Continuous probability distributions

• 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

• Because the total area under the density curve equals one, there is correspondence between area and probability

Density Curve

A factory produces a part with measurement of length[50cm, 60cm]. An inspector randomly measured 100parts produced and created  relative frequency-based probability histogram where length of parts is grouped every 1cm. 

Example

If the inspector randomly measured 200parts produced and created a probability histogram where length of parts is grouped every 0.5 cm.

If assume that the inspector measured unlimited parts and the class interval for  relative frequency/ probability became extremely small, for instance, is only able to accommodate one possible value.

• The bars in histogram will become adjacent lines.

• the top of the lines will be connected as a curve.

1. A continuous random variable has a uniform distribution if its values spread evenly over the range of possibilities.

2. The probability such that the values located within all regions/intervals of the same length on the range of possibilities are equally probable.

3. A uniform continuous distribution must have two parameters, the minimum possible value (α) and the maximum possible value( b).

   1. The density curve is a straight line.  The probability area will be a rectangular-shaped graph

Uniform Probability density function is :where a is the smallest value the variable can assume, b is the largest value the variable can assume.

Cumulative distribution function (CDF)

Expected value of x 

Variance of x

If a continuous random variable has a distribution with a graph that is symmetric and bell-shaped 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.

Mean equal to zero, Standard deviation equal to one which is N(0,1).

A non-normal distribution can be converted to a standard normal distribution of its z-score values;

The transformation of from non-normal to normal distributions shown on the graph below;