About distribution of random variables
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I'm understand what is $mathrm P$ ($eta leq x)$ for a random variable, but what is $mathrm P$ ($eta leq xi$) if $xi$ is random variable too? I can't find definition of something like that. How to calculate it, if I know that $eta, xi$ are independent and have the same geometric distribution with the parameter $p$. Will it be some function from $p$?
probability probability-distributions random-variables
asked Nov 21 at 15:37
anykk
665
add a comment |
up vote
2
down vote
favorite
I'm understand what is $mathrm P$ ($eta leq x)$ for a random variable, but what is $mathrm P$ ($eta leq xi$) if $xi$ is random variable too? I can't find definition of something like that. How to calculate it, if I know that $eta, xi$ are independent and have the same geometric distribution with the parameter $p$. Will it be some function from $p$?
probability probability-distributions random-variables
asked Nov 21 at 15:37
anykk
665
add a comment |
up vote
2
down vote
favorite
up vote
2
down vote
favorite
I'm understand what is $mathrm P$ ($eta leq x)$ for a random variable, but what is $mathrm P$ ($eta leq xi$) if $xi$ is random variable too? I can't find definition of something like that. How to calculate it, if I know that $eta, xi$ are independent and have the same geometric distribution with the parameter $p$. Will it be some function from $p$?
probability probability-distributions random-variables
asked Nov 21 at 15:37
anykk
665
I'm understand what is $mathrm P$ ($eta leq x)$ for a random variable, but what is $mathrm P$ ($eta leq xi$) if $xi$ is random variable too? I can't find definition of something like that. How to calculate it, if I know that $eta, xi$ are independent and have the same geometric distribution with the parameter $p$. Will it be some function from $p$?
probability probability-distributions random-variables
probability probability-distributions random-variables
asked Nov 21 at 15:37
anykk
665
asked Nov 21 at 15:37
anykk
665
edited Nov 21 at 15:47
asked Nov 21 at 15:37
anykk
665
asked Nov 21 at 15:37
anykk
665
asked Nov 21 at 15:37
anykk
665
665
add a comment |
add a comment |
1 Answer
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If $eta$ and $xi$ are both random variables on the same probability space, then also $eta-xi$ is a random variable on that probability space.
Now observe that $${etaleqxi}={eta-xileq0}$$ so that $$P({etaleqxi})=P({eta-xileq0})$$
One way to find (mostly not the most convenient one) is finding the distribution of $eta-xi$.
Also it can be found as $$mathbb E[etaleqxi]=intint[xleq y]dF_{eta,xi}(x,y)tag1$$ where $[xleq y]$ denotes the function $mathbb R^2tomathbb R$ that takes value $1$ if $xleq y$ and takes value $0$ otherwise, and $F_{eta,xi}$ denotes the CDF of random vector $(eta,xi)$.
If $eta$ and $xi$ are independent then the RHS of $(1)$ becomes:$$int_{-infty}^{infty}int_{-infty}^ydF_{eta}(x)dF_{xi}(y)=int_{-infty}^{infty}F_{eta}(y)dF_{xi}(y)$$
answered Nov 21 at 15:45
drhab
96.3k543126
Eh, you have helped me twice, thank you. Could you recommend some lectures or books about probability theory and statistics? I'm from Russia and unfamiliar with english literature, but I think, that it'll better then most that I know
– anykk
Nov 21 at 15:52
Sorry, but I learned prob. and stat. not from books but from scripts that were used at university. So I am not quite familiar with english books on that stuff either, and cannot recommend something.
– drhab
Nov 21 at 15:58
add a comment |
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1 Answer
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oldest
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1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
accepted
If $eta$ and $xi$ are both random variables on the same probability space, then also $eta-xi$ is a random variable on that probability space.
Now observe that $${etaleqxi}={eta-xileq0}$$ so that $$P({etaleqxi})=P({eta-xileq0})$$
One way to find (mostly not the most convenient one) is finding the distribution of $eta-xi$.
Also it can be found as $$mathbb E[etaleqxi]=intint[xleq y]dF_{eta,xi}(x,y)tag1$$ where $[xleq y]$ denotes the function $mathbb R^2tomathbb R$ that takes value $1$ if $xleq y$ and takes value $0$ otherwise, and $F_{eta,xi}$ denotes the CDF of random vector $(eta,xi)$.
If $eta$ and $xi$ are independent then the RHS of $(1)$ becomes:$$int_{-infty}^{infty}int_{-infty}^ydF_{eta}(x)dF_{xi}(y)=int_{-infty}^{infty}F_{eta}(y)dF_{xi}(y)$$
answered Nov 21 at 15:45
drhab
96.3k543126
Eh, you have helped me twice, thank you. Could you recommend some lectures or books about probability theory and statistics? I'm from Russia and unfamiliar with english literature, but I think, that it'll better then most that I know
– anykk
Nov 21 at 15:52
Sorry, but I learned prob. and stat. not from books but from scripts that were used at university. So I am not quite familiar with english books on that stuff either, and cannot recommend something.
– drhab
Nov 21 at 15:58
add a comment |
up vote
2
down vote
accepted
If $eta$ and $xi$ are both random variables on the same probability space, then also $eta-xi$ is a random variable on that probability space.
Now observe that $${etaleqxi}={eta-xileq0}$$ so that $$P({etaleqxi})=P({eta-xileq0})$$
One way to find (mostly not the most convenient one) is finding the distribution of $eta-xi$.
Also it can be found as $$mathbb E[etaleqxi]=intint[xleq y]dF_{eta,xi}(x,y)tag1$$ where $[xleq y]$ denotes the function $mathbb R^2tomathbb R$ that takes value $1$ if $xleq y$ and takes value $0$ otherwise, and $F_{eta,xi}$ denotes the CDF of random vector $(eta,xi)$.
If $eta$ and $xi$ are independent then the RHS of $(1)$ becomes:$$int_{-infty}^{infty}int_{-infty}^ydF_{eta}(x)dF_{xi}(y)=int_{-infty}^{infty}F_{eta}(y)dF_{xi}(y)$$
answered Nov 21 at 15:45
drhab
96.3k543126
Eh, you have helped me twice, thank you. Could you recommend some lectures or books about probability theory and statistics? I'm from Russia and unfamiliar with english literature, but I think, that it'll better then most that I know
– anykk
Nov 21 at 15:52
Sorry, but I learned prob. and stat. not from books but from scripts that were used at university. So I am not quite familiar with english books on that stuff either, and cannot recommend something.
– drhab
Nov 21 at 15:58
add a comment |
up vote
2
down vote
accepted
up vote
2
down vote
accepted
If $eta$ and $xi$ are both random variables on the same probability space, then also $eta-xi$ is a random variable on that probability space.
Now observe that $${etaleqxi}={eta-xileq0}$$ so that $$P({etaleqxi})=P({eta-xileq0})$$
One way to find (mostly not the most convenient one) is finding the distribution of $eta-xi$.
Also it can be found as $$mathbb E[etaleqxi]=intint[xleq y]dF_{eta,xi}(x,y)tag1$$ where $[xleq y]$ denotes the function $mathbb R^2tomathbb R$ that takes value $1$ if $xleq y$ and takes value $0$ otherwise, and $F_{eta,xi}$ denotes the CDF of random vector $(eta,xi)$.
If $eta$ and $xi$ are independent then the RHS of $(1)$ becomes:$$int_{-infty}^{infty}int_{-infty}^ydF_{eta}(x)dF_{xi}(y)=int_{-infty}^{infty}F_{eta}(y)dF_{xi}(y)$$
answered Nov 21 at 15:45
drhab
96.3k543126
If $eta$ and $xi$ are both random variables on the same probability space, then also $eta-xi$ is a random variable on that probability space.
Now observe that $${etaleqxi}={eta-xileq0}$$ so that $$P({etaleqxi})=P({eta-xileq0})$$
One way to find (mostly not the most convenient one) is finding the distribution of $eta-xi$.
Also it can be found as $$mathbb E[etaleqxi]=intint[xleq y]dF_{eta,xi}(x,y)tag1$$ where $[xleq y]$ denotes the function $mathbb R^2tomathbb R$ that takes value $1$ if $xleq y$ and takes value $0$ otherwise, and $F_{eta,xi}$ denotes the CDF of random vector $(eta,xi)$.
If $eta$ and $xi$ are independent then the RHS of $(1)$ becomes:$$int_{-infty}^{infty}int_{-infty}^ydF_{eta}(x)dF_{xi}(y)=int_{-infty}^{infty}F_{eta}(y)dF_{xi}(y)$$
answered Nov 21 at 15:45
drhab
96.3k543126
edited Nov 21 at 16:01
answered Nov 21 at 15:45
drhab
96.3k543126
answered Nov 21 at 15:45
drhab
96.3k543126
answered Nov 21 at 15:45
drhab
96.3k543126
96.3k543126
Eh, you have helped me twice, thank you. Could you recommend some lectures or books about probability theory and statistics? I'm from Russia and unfamiliar with english literature, but I think, that it'll better then most that I know
– anykk
Nov 21 at 15:52
Sorry, but I learned prob. and stat. not from books but from scripts that were used at university. So I am not quite familiar with english books on that stuff either, and cannot recommend something.
– drhab
Nov 21 at 15:58
add a comment |
Eh, you have helped me twice, thank you. Could you recommend some lectures or books about probability theory and statistics? I'm from Russia and unfamiliar with english literature, but I think, that it'll better then most that I know
– anykk
Nov 21 at 15:52
Sorry, but I learned prob. and stat. not from books but from scripts that were used at university. So I am not quite familiar with english books on that stuff either, and cannot recommend something.
– drhab
Nov 21 at 15:58
Eh, you have helped me twice, thank you. Could you recommend some lectures or books about probability theory and statistics? I'm from Russia and unfamiliar with english literature, but I think, that it'll better then most that I know
– anykk
Nov 21 at 15:52
Eh, you have helped me twice, thank you. Could you recommend some lectures or books about probability theory and statistics? I'm from Russia and unfamiliar with english literature, but I think, that it'll better then most that I know
– anykk
Nov 21 at 15:52
Sorry, but I learned prob. and stat. not from books but from scripts that were used at university. So I am not quite familiar with english books on that stuff either, and cannot recommend something.
– drhab
Nov 21 at 15:58
Sorry, but I learned prob. and stat. not from books but from scripts that were used at university. So I am not quite familiar with english books on that stuff either, and cannot recommend something.
– drhab
Nov 21 at 15:58
add a comment |
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