CDF's properties
I need a help to prove all the properties of CDF to better understand properties and characteristics of bidimensional CDF.
By definition of CDF, I know that:
$F_X:mathbb{R}rightarrow [0,1]: F_X(x)=mathbb{P}(Xleq x)=left{begin{matrix}
sum_{x_ileq x}p_X(x_i) & discrete\
int_{_-infty}^{x}f_X(x)dx & continuous
end{matrix}right.
$
1) $0leq F_X(x)leq 1$ (I don't think that there's nothing to prove, given the domain $[0,1]$).
2) $lim_{xrightarrow -infty}F_X(x)=0$: I don't know how to prove it. My text is really dismissive about that. It says only (I quote) that "The property is obvious. If the argument of $F_X$ goes to $-infty$, the reference set goes to $varnothing $ and the probability is zero".
And for the bidimensional CDF it says that "...if one of to arguments of CDF goes to $-infty$..."
$lim_{X_1rightarrow-infty }F_{X_1,X_2}(x_1,x_2)=lim_{X_2rightarrow-infty }F_{X_1,X_2}(x_1,x_2)=0$.
3) $lim_{xrightarrow +infty}F_X(x)=1$: The same of above. I can only say that, being a cumulative function, the probability ranged in $(a,+infty)$ must be $=1$.
And again, for the bidimensional CDF it says only that "to obtain the marginal CDFs, it's enough to converg the other argument to $+infty$ for having a reference set that coincides with a half-line...
$lim_{X_1rightarrow+infty }F_{X_1,X_2}(x_1,x_2)=F_{X_2}(x_2)$
$lim_{X_1,X_2rightarrow+infty }F_{X_1,X_2}(x_1,x_2)=1$
4) $F$ is a monotonic function: It says nothing about that. But then, for bidimensional CDF, this property has shown to be fundamental to prove that exist an (alleged) link between CDF and the value of probability o rectangles of $mathbb{R}^2$.
$mathbb{P}(a_1<X_1<b_1,a_2<X_2<b_2)=Delta _{a_1b_1}^{a_2b_2}F(X_1,X_2)$
One formula that rained down from the sky...like that.
I'm sorry for this question. I'm so bummed: it's impossible to study with a similar text.
Thanks for any help.
probability multivariable-calculus random-variables density-function
asked Dec 13 '18 at 10:27
Marco PittellaMarco Pittella
1308
add a comment |
I need a help to prove all the properties of CDF to better understand properties and characteristics of bidimensional CDF.
By definition of CDF, I know that:
$F_X:mathbb{R}rightarrow [0,1]: F_X(x)=mathbb{P}(Xleq x)=left{begin{matrix}
sum_{x_ileq x}p_X(x_i) & discrete\
int_{_-infty}^{x}f_X(x)dx & continuous
end{matrix}right.
$
1) $0leq F_X(x)leq 1$ (I don't think that there's nothing to prove, given the domain $[0,1]$).
2) $lim_{xrightarrow -infty}F_X(x)=0$: I don't know how to prove it. My text is really dismissive about that. It says only (I quote) that "The property is obvious. If the argument of $F_X$ goes to $-infty$, the reference set goes to $varnothing $ and the probability is zero".
And for the bidimensional CDF it says that "...if one of to arguments of CDF goes to $-infty$..."
$lim_{X_1rightarrow-infty }F_{X_1,X_2}(x_1,x_2)=lim_{X_2rightarrow-infty }F_{X_1,X_2}(x_1,x_2)=0$.
3) $lim_{xrightarrow +infty}F_X(x)=1$: The same of above. I can only say that, being a cumulative function, the probability ranged in $(a,+infty)$ must be $=1$.
And again, for the bidimensional CDF it says only that "to obtain the marginal CDFs, it's enough to converg the other argument to $+infty$ for having a reference set that coincides with a half-line...
$lim_{X_1rightarrow+infty }F_{X_1,X_2}(x_1,x_2)=F_{X_2}(x_2)$
$lim_{X_1,X_2rightarrow+infty }F_{X_1,X_2}(x_1,x_2)=1$
4) $F$ is a monotonic function: It says nothing about that. But then, for bidimensional CDF, this property has shown to be fundamental to prove that exist an (alleged) link between CDF and the value of probability o rectangles of $mathbb{R}^2$.
$mathbb{P}(a_1<X_1<b_1,a_2<X_2<b_2)=Delta _{a_1b_1}^{a_2b_2}F(X_1,X_2)$
One formula that rained down from the sky...like that.
I'm sorry for this question. I'm so bummed: it's impossible to study with a similar text.
Thanks for any help.
probability multivariable-calculus random-variables density-function
asked Dec 13 '18 at 10:27
Marco PittellaMarco Pittella
1308
add a comment |
I need a help to prove all the properties of CDF to better understand properties and characteristics of bidimensional CDF.
By definition of CDF, I know that:
$F_X:mathbb{R}rightarrow [0,1]: F_X(x)=mathbb{P}(Xleq x)=left{begin{matrix}
sum_{x_ileq x}p_X(x_i) & discrete\
int_{_-infty}^{x}f_X(x)dx & continuous
end{matrix}right.
$
1) $0leq F_X(x)leq 1$ (I don't think that there's nothing to prove, given the domain $[0,1]$).
2) $lim_{xrightarrow -infty}F_X(x)=0$: I don't know how to prove it. My text is really dismissive about that. It says only (I quote) that "The property is obvious. If the argument of $F_X$ goes to $-infty$, the reference set goes to $varnothing $ and the probability is zero".
And for the bidimensional CDF it says that "...if one of to arguments of CDF goes to $-infty$..."
$lim_{X_1rightarrow-infty }F_{X_1,X_2}(x_1,x_2)=lim_{X_2rightarrow-infty }F_{X_1,X_2}(x_1,x_2)=0$.
3) $lim_{xrightarrow +infty}F_X(x)=1$: The same of above. I can only say that, being a cumulative function, the probability ranged in $(a,+infty)$ must be $=1$.
And again, for the bidimensional CDF it says only that "to obtain the marginal CDFs, it's enough to converg the other argument to $+infty$ for having a reference set that coincides with a half-line...
$lim_{X_1rightarrow+infty }F_{X_1,X_2}(x_1,x_2)=F_{X_2}(x_2)$
$lim_{X_1,X_2rightarrow+infty }F_{X_1,X_2}(x_1,x_2)=1$
4) $F$ is a monotonic function: It says nothing about that. But then, for bidimensional CDF, this property has shown to be fundamental to prove that exist an (alleged) link between CDF and the value of probability o rectangles of $mathbb{R}^2$.
$mathbb{P}(a_1<X_1<b_1,a_2<X_2<b_2)=Delta _{a_1b_1}^{a_2b_2}F(X_1,X_2)$
One formula that rained down from the sky...like that.
I'm sorry for this question. I'm so bummed: it's impossible to study with a similar text.
Thanks for any help.
probability multivariable-calculus random-variables density-function
asked Dec 13 '18 at 10:27
Marco PittellaMarco Pittella
1308
I need a help to prove all the properties of CDF to better understand properties and characteristics of bidimensional CDF.
By definition of CDF, I know that:
$F_X:mathbb{R}rightarrow [0,1]: F_X(x)=mathbb{P}(Xleq x)=left{begin{matrix}
sum_{x_ileq x}p_X(x_i) & discrete\
int_{_-infty}^{x}f_X(x)dx & continuous
end{matrix}right.
$
1) $0leq F_X(x)leq 1$ (I don't think that there's nothing to prove, given the domain $[0,1]$).
2) $lim_{xrightarrow -infty}F_X(x)=0$: I don't know how to prove it. My text is really dismissive about that. It says only (I quote) that "The property is obvious. If the argument of $F_X$ goes to $-infty$, the reference set goes to $varnothing $ and the probability is zero".
And for the bidimensional CDF it says that "...if one of to arguments of CDF goes to $-infty$..."
$lim_{X_1rightarrow-infty }F_{X_1,X_2}(x_1,x_2)=lim_{X_2rightarrow-infty }F_{X_1,X_2}(x_1,x_2)=0$.
3) $lim_{xrightarrow +infty}F_X(x)=1$: The same of above. I can only say that, being a cumulative function, the probability ranged in $(a,+infty)$ must be $=1$.
And again, for the bidimensional CDF it says only that "to obtain the marginal CDFs, it's enough to converg the other argument to $+infty$ for having a reference set that coincides with a half-line...
$lim_{X_1rightarrow+infty }F_{X_1,X_2}(x_1,x_2)=F_{X_2}(x_2)$
$lim_{X_1,X_2rightarrow+infty }F_{X_1,X_2}(x_1,x_2)=1$
4) $F$ is a monotonic function: It says nothing about that. But then, for bidimensional CDF, this property has shown to be fundamental to prove that exist an (alleged) link between CDF and the value of probability o rectangles of $mathbb{R}^2$.
$mathbb{P}(a_1<X_1<b_1,a_2<X_2<b_2)=Delta _{a_1b_1}^{a_2b_2}F(X_1,X_2)$
One formula that rained down from the sky...like that.
I'm sorry for this question. I'm so bummed: it's impossible to study with a similar text.
Thanks for any help.
probability multivariable-calculus random-variables density-function
probability multivariable-calculus random-variables density-function
asked Dec 13 '18 at 10:27
Marco PittellaMarco Pittella
1308
asked Dec 13 '18 at 10:27
Marco PittellaMarco Pittella
1308
asked Dec 13 '18 at 10:27
Marco PittellaMarco Pittella
1308
asked Dec 13 '18 at 10:27
Marco PittellaMarco Pittella
1308
asked Dec 13 '18 at 10:27
Marco PittellaMarco Pittella
1308
1308
add a comment |
add a comment |
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