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# MATH 10C Chapter Notes -Probability Density Function, Antiderivative

Page:
of 2 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)