# Uniform Cumulative Distribution

Cumulative distribution
function (fsk) or more concisely the distribution function F for random
variable X is defined as

*F(b)=P{X≤b}*

or all real numbers

*b, ∞ <b <∞*. In words,*F (b)*states the chance that random variable*X*takes a value smaller or equal to*b*. Some properties of f.s.k*F*are
1. F is a non-encasing function,
meaning that if a <b, then

3. *F (a) ≤F (b)*.4. F (b) continuous right, meaning

In
this case lim means the limit for under the condition that
every .Character 1 is a result of the fact that if a < b, then the
event {X ≤ a} is included in the event {X ≤ b}, so that it cannot have a
greater. Properties 2, 3, and 4 are all a result of the continuity of
probability. For example, to prove property 2, note that if , then
events converge to event {X <∞} (meaning, if , then lim. Therefore, based on the nature of continuity of opportunity,
we get.

Evidence
for properties is similar to the above and is provided as an exercise. To prove
Nature 4, note that , then . This is a result of the
fact that for infinite number of n if and only if {X ≤ b} (so, ), likewise, for all except for the
number n if and only if X≤b (so,) . So and the continuity
character produces,

or

So that is proven Nature 4.

All kinds of opportunity
questions about X can be answered based on f.s.k F. For example,

*P {a <X ≤ b} = F (b) -F (a)*for all

*a < b*.................. equation 1

This
is most easily seen if the {X ≤ b} event is pronounced as a combination of events
{X ≤ a} and {a < X ≤ b} that set aside each other. In other words.

So that

Which proves equation 1.

If we want to calculate the chance that

*X*is smaller than*b*, we can also apply the continuity to obtain,
Note that

*P {X < b}*is not always equal to*F(b)*, because*F(b)*also includes the chance that*X*is equal to*b*.
Suppose the function of the random variable X is known as follows,

Calculate (a) P{X<3}, (b) P{X=1},
(c) P{X>1/2}, dan (d) P{2<X≤4}.

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