From discrete to continuous

Continuous variables and probability density

So far random variables took isolated values: 0, 1, 2 successes. But a height, a duration, a temperature can take any value in an interval. A different tool is then needed.

The problem with the continuum

What is the probability that a person is exactly 1.750000... m tall? Zero. Between 1.74 and 1.76 there are infinitely many possible values; none can have non-zero probability, or the total would exceed 1.

DISCRETE variable                  CONTINUOUS variable
------------------------------     ------------------------------
P(X = 3) is meaningful             P(X = 1.75) = 0 always
probabilities are ADDED            we speak of INTERVALS
bar chart                          continuous curve

A continuous variable is therefore described only through intervals: P(1.74 ≤ X ≤ 1.76).

Practical consequence: strict and non-strict inequalities give the same result.

P(X ≤ a) = P(X < a)          since P(X = a) = 0

The probability density

The bar chart is replaced by a density curve f. Probability becomes an area under the curve:

   f(x)
     |        ___
     |      _/###\_              P(a ≤ X ≤ b) = shaded area
     |    _/ #####  \_
     |  _/   #####    \_
     +---|---#####---|-------> x
         a           b

Two rules define a density:

1. f(x) ≥ 0 everywhere                  (no negative area)
2. the TOTAL area under the curve is 1  (something must happen)

This is exactly the continuous transposition of "the probabilities add to 1".

From histogram to curve

The transition is easy to picture. Refining the classes of a histogram of heights:

10 cm classes          2 cm classes            infinitely fine classes
    ▄▆█▆▄                ▁▃▅▇█▇▅▃▁                    ___
   ▄▆███▆▄              ▁▃▅▇███▇▅▃▁               __/   \__
  ------------          --------------          --------------

The frequency histogram smooths into a curve: the density. A bar's area was a frequency; the area under the curve becomes a probability.

Expectation and standard deviation

The notions are unchanged, the sum becoming an integral:

E(X) = ∫ x f(x) dx           the abscissa of the area's "centre of mass"
V(X) = E(X²) - [E(X)]²       the spread about that centre

They are interpreted as before: E(X) locates the centre, σ(X) measures the spread.

Summary