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).
3. ![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgFxpDTYa0c1a3JKP6b3nnGn0ZzJxb2BkTEEp_XGwkyXfBY1QyFcXndQZ7nWo2H4H_IjxoAQ7mpPs8F6iicFwttI4IaIifeZUdGUBPLTfBZ9RFDBxtp4B-uyg60S1I30ZfmqvVGIZAenYI/s1600/image003.png)
4. F (b) continuous right, meaning
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgz1lR3MbAc55AV3R4cmBhFNM6y0XiCbFbRRCfxfsNtc2CtGq8-KC0_EhR5u6FXnm_IH49QScxAChU_ea0SIQu3iIF3ySyv5E_Ek2kr487DM1DhXizW4_3HnNwHMfsZLMTp5PcnMNk83BY/s1600/image005.png)
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.
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjpoSXZUk9sRvjWL9iJsL5tAkTgMx-pvdG6L9LuihYW_DuwFiG4F0h64pvbyK0GpDQD-xF88X5e6DTZaKjamu8jiZwVM2qGfmV8l74coo_IVPO62fIGVFMNFC6Pj-7j-wajuWpa41D9xUU/s1600/image007.png)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjnnF-pKnVm9q9hA3aMDEEkwNfR2i_VUpfMxIyLzGBhmKrzIfLdqXpzBSBlYEjoqYm515KYDuCTmKJt9w7LQomzzdfmOIpX4IEJW208Nc0PtKuPN6Wm0nq4VFvH62QuV-ksm8-GkwRLlpM/s1600/image010.png)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEivB6V7gK6GtStlJFuA3M7YXoD87n06C-ronSHHb-k8_Uz_lG3o5ArIQWeRlNX5xtqlkvtDlrKgvRpuMkg-QhIcYS00H7PNvxyL_T6_dPFYvgvSqtcWTuah8w7CzUHrwjjOGELIySlZ6sc/s1600/image012.png)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh7tZw-S9Zo3WNMYYs2x4KucCLJyyaYF6BNb3d1r2u98jnGWjW2Mvtwvk4ldNhenyDliBqHoCGYYT1sYHxFiHq-0z2i5VXTvwyDqXrQw-EYvN_LOKCxsUNOYSN1dmVC0YGP6FKyuDy-PAc/s1600/image013.png)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEivB6V7gK6GtStlJFuA3M7YXoD87n06C-ronSHHb-k8_Uz_lG3o5ArIQWeRlNX5xtqlkvtDlrKgvRpuMkg-QhIcYS00H7PNvxyL_T6_dPFYvgvSqtcWTuah8w7CzUHrwjjOGELIySlZ6sc/s1600/image012.png)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiCQtEn4t1SwIRXjTPniraDpnouwccWofTWkIxzhIojS4iAahrklbu7BE05pY4u5he5Z6PxEa1HLe1FeVlAKhw1aLR0lPxbQGloZIJLKotexjAPldD5rxRTAtRnDvtv1xhQ91ugd9NaL3U/s1600/image018.png)
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,
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg0iDOc_6PZMLSnVLRxTFK87qKOFSUBFxOxHxMdYmsqVM2pkKhzi4Zpqx3tQ3gpF6hFVCChNClO8mWlqkTcGbYX9skJV5bhr2q7x0QJEmb3ZzdAcZLW8zf6o2SAsT4okNcGCgaxgXY4D8g/s1600/image021.png)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiGXwcQdKVXZRnU4_F4iS4sLBTtVqcJvgXR8q9KAN8JKc3gkMcZd_DvFw3tWVJWtV3kaQm770EdUUy2tHVIXCR4_uC6TIm-TiWtUqGQaeIYN_hfT3guvWrzibguwFg4NIDm1SQ6P4Qev_U/s1600/image017.png)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjGb5nDmRNrKX_o8b90MyRig2aCeMkszC846smzjW9H8HUdYiIBRxTrC3y06T9GZ8k9aa8grfQb8wApw1R-3yYZgsADgMccjzMadTyP_bJvHTZx66xsAaG9kHsHTYm6rnmgJC4pSVvhlns/s1600/image014.png)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgOZ10we5d4aVKJ6HWZCMIGJH3Z1WVTXGSWOTqqiE29rOvaNv2NFFxM1Ad-UrcEvHsNfAbQYLzcAg1LLT2rqi8y0WUW35nxgFrU_-2Q45Ct3bm_3pNpnTXSI_0KXguJ2GyHMM31cFZjFJE/s1600/image027.png)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjj3Z3P9Xp2EHZmzlA0uL28-uVCemeMtW4x8hfBJFK5wlPbVsdC-OlEt25A_W4Eq5uk77D9MrKw7mDJMQW0N7_3PBefZU4HvE5sJz1nCcxIkSCVpcWMEaxhkzrz_LtC9MKoryx3dsZzUc4/s1600/image026.png)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg0GKICaqHQoZtEEnzQKljqzy85oRPFQ-RVjZmt2uUd3hLTqOYgrFtIttMF7eISCU4ZjWnT9iBJ0tY1NSDLODQc9Cqob_6IMyBtYEEQ62La5RFTbzu5Re1_TCk3laLEcngp7i8wFgSSpxg/s1600/image031.png)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhm-wOWN-5L5I01Dmi3WENqB9DK8IZnU25TggYDDahzzkERl6gBQkZf91r5WSHGkJNKWNQ-boak9ldhap8xYxau5RLJjhQjJhoGLzaVamEIq8MPrQOeIwDu_SWXfyR80Vvvx6UYskDvadQ/s1600/image078.png)
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}.
SUBSCRIBE TO OUR NEWSLETTER
0 Response to " Uniform Cumulative Distribution"
Post a Comment