Concept:

How to represent distribution of various quantities

through the population using a density function

Histogram

Place a vertical bar above each category

The area of each bar represents the franction of the

population in that category

Total are is 100% = 1

Assume there is an even distribution under the bar

This may be an under or over estimation for some

fractions within the category

More categories = smoother histogram = better

estimation

The Density function

A function that “smooths out” the histogram

Fraction of population between a and b= Area under

graph of between a and b = ab p(t) dt

-∞∞ p(x)dx =1 an p(x) ≥ 0 for all x

Cumulative Distribution Function

P(t) = -∞ ∞ p(x) dx = Fraction of population having

values of x below t

Thus, P is an antiderivative of p, that is P’(x) = p

Any cumulative distribtution has the following

properties:

P is increasing (or non decreasing)

Lim P(t) =1 and Lim P(t) =0

t-> ∞ t->∞

Fraction of population having values of x between

a and b

ba p(x) dx = P(b) –P(a)

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