VARIATION AND PROBABILITY

CONTINUOUS RANDOM VARIABLES

These are used to define probability models for continuous scale measurements,

Relative Frequency Histogram

For a large data set we summarise the distribution using a relative frequency histogram

where the relative frequency of observations between a and b is proportional to the areas of the rectangles above [a,b].

As sample size increases

Probability Density Function

So for a continuous random variable X, we describe the probability distribution by some function f(x) e.g.

such that

(i) f(x) >= 0 for all x

(ii) area under the curve between a and b is

(iii) Total area under curve = 1.

f(x) is called the probability density function of X.

For a continuous random variable the probability of it taking a particular value exactly, e.g. X = length of a bolt = 1.999965722 cms, is zero. That is

Instead for continuous random variables probabilities are associated with a range of values.

e.g. 1.95 <= X <= 2.00 cms.

Example - Uniform Continuous Distribution

One example of a probability density function for a continuous random variable is the uniform continuous distribution.

X can take any real value between a and b with probability uniform over this interval.

Total area = 1 = length x height

Thus the probability density function is

For any values c and d between a and b

Uniform continuous random number generation in MINITAB

Random numbers with this distribution can be generated by MINITAB, e.g. using a = 0 and b = 1

          RANDOM   5   C1;
          UNIFORM   0   1 .
          PRINT   C1

gave

0.270022        0.121774        0.835736 ...