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density function∗
drini†
2013-03-21 14:53:00
Let X be a discrete random variable with sample space {x1 , x2 , . . .}. Let pk
be the probability of X taking the value xk .
The function
(
pk if x = xk
f (x) =
0 otherwise
is called the probability function or density function.
It must hold:
∞
X
f (xj ) = 1
j=1
If the density function for a random variable is known, we can calculate the
probability of X being on certain interval:
X
X
P [a < X ≤ b] =
f (xj ) =
pj .
a<xj ≤b
a<xj ≤b
The definition can be extended to continuous random variables in a direct
way: The probability of x being on a given interval is calculated with an integral
instead of using a summation:
Z
P [a < X ≤ b] =
b
f (x)dx.
a
For a more formal approach using measure theory, look at probability distribution function entry.
∗ hDensityFunctioni
created: h2013-03-21i by: hdrinii version: h33450i Privacy setting:
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