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# 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 F (a) ≤F (b).

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}.