Cumulative distribution function: Difference between revisions

(Getting rid of this idiotic phrase, which is not the correct title of the linked page.)
Example of joint cumulative distribution function:
 
For two continuous variables ''X'' and ''Y'': P(<math> \Pr(a<X<b) \text{ and( } c<Y<d)) =<math> \int\limits_{a}limits_a^{b} \int\limits_{c}limits_c^{d} f(x,y) \, dy \, dx</math>;
 
For two discrete random variables, it is beneficial to generate a table of probabilities and address the cumulative probability for each potential range of ''X'' and ''Y'', and here is the example<ref>{{Cite web|url=https://math.info/Probability/Joint_CDF/|title=Joint Cumulative Distribution Function (CDF)|website=math.info|access-date=2019-12-11}}</ref>:
 
given the joint probability density function in tabular form, determine the joint cumulative distribution function.
{| class="wikitable"
|
|''Y'' = 2
|''Y'' = 4
|''Y'' = 6
|''Y'' = 8
|-
|''X'' = 1
|0
|0.1
|0.1
|-
|''X'' = 3
|0
|0
|0
|-
|''X'' = 5
|0.3
|0
|0.15
|-
|''X'' = 7
|0
|0
|0
|}
Solution: using the given table of probabilities for each potential range of ''X'' and ''Y'', the joint cumulative distribution function may be constructed in tabular form:
{| class="wikitable"
|
|''Y'' < 2
|2 ≤ ''Y'' < 4
|2≤Y<4
|4 ≤ ''Y'' < 6
|4≤Y<6
|6 ≤ ''Y'' < 8
|6≤Y<8
|''Y'' ≤ 8
|Y≤8
|-
|''X'' < 1
|0
|0
|0
|-
|1 ≤ ''X'' < 3
|1≤X<3
|0
|0
|0.2
|-
|3 ≤ ''X'' < 5
|3≤X<5
|0
|0
|0.4
|-
|5 ≤ ''X'' < 7
|5≤X<7
|0
|0.3
|0.85
|-
|''X'' ≤ 7
|X≤7
|0
|0.3
|}
<br />
 
===Definition for more than two random variables===
For <math>N</math> random variables <math>X_1,\ldots,X_N</math>, the joint CDF <math>F_{X_1,\ldots,X_N}</math> is given by