How to calculate the probability of x is bigger than y while they have the same distribution but different mean values?

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I am reading an engineering book. The authors mentioned that variables $x,y$ are exponentially distributed with a mean $a^2+N_o$ and $N_o$, respectively. We can calculate the probability of error by direction integration, that is:
$$p_e=mathbbPy>x=[2+fraca^2N_o]^-1$$
Where the probability density function of the r.v. is $f(u)=frac1mucdotexp^frac-umu$ with $mu$ is the mean. My question is how to calculate the error probability since $x$ and $y$ have different mean values?
To my knowledge, I think we can calculate $mathbbPy>0$ with $int_0^infty
frac1N_ocdotexp^frac-uN_odu$ (please correct me if I am wrong). Thus, how do I extend it to two r.v. with different mean values?




probability-distributions density-function




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  • I don't understand -- you say "We can calculate ..." and give the result, but then you seem to ask how to obtain it. Is this a result from the book? Who's the "we" who can calculate it?
    – joriki
    Jul 22 at 10:41











  • I think this post will answer your question
    – David M.
    Jul 22 at 15:34










  • Thanks for the help guys. I have found a relevant link. I apologize for the repeated question. The answer can be found here: math.stackexchange.com/questions/1805587/…
    – M.A.N
    Jul 23 at 1:02














up vote
0
down vote

favorite












I am reading an engineering book. The authors mentioned that variables $x,y$ are exponentially distributed with a mean $a^2+N_o$ and $N_o$, respectively. We can calculate the probability of error by direction integration, that is:
$$p_e=mathbbPy>x=[2+fraca^2N_o]^-1$$
Where the probability density function of the r.v. is $f(u)=frac1mucdotexp^frac-umu$ with $mu$ is the mean. My question is how to calculate the error probability since $x$ and $y$ have different mean values?
To my knowledge, I think we can calculate $mathbbPy>0$ with $int_0^infty
frac1N_ocdotexp^frac-uN_odu$ (please correct me if I am wrong). Thus, how do I extend it to two r.v. with different mean values?




probability-distributions density-function




share|cite|improve this question





















  • I don't understand -- you say "We can calculate ..." and give the result, but then you seem to ask how to obtain it. Is this a result from the book? Who's the "we" who can calculate it?
    – joriki
    Jul 22 at 10:41











  • I think this post will answer your question
    – David M.
    Jul 22 at 15:34










  • Thanks for the help guys. I have found a relevant link. I apologize for the repeated question. The answer can be found here: math.stackexchange.com/questions/1805587/…
    – M.A.N
    Jul 23 at 1:02












up vote
0
down vote

favorite









up vote
0
down vote

favorite











I am reading an engineering book. The authors mentioned that variables $x,y$ are exponentially distributed with a mean $a^2+N_o$ and $N_o$, respectively. We can calculate the probability of error by direction integration, that is:
$$p_e=mathbbPy>x=[2+fraca^2N_o]^-1$$
Where the probability density function of the r.v. is $f(u)=frac1mucdotexp^frac-umu$ with $mu$ is the mean. My question is how to calculate the error probability since $x$ and $y$ have different mean values?
To my knowledge, I think we can calculate $mathbbPy>0$ with $int_0^infty
frac1N_ocdotexp^frac-uN_odu$ (please correct me if I am wrong). Thus, how do I extend it to two r.v. with different mean values?




probability-distributions density-function




share|cite|improve this question













I am reading an engineering book. The authors mentioned that variables $x,y$ are exponentially distributed with a mean $a^2+N_o$ and $N_o$, respectively. We can calculate the probability of error by direction integration, that is:
$$p_e=mathbbPy>x=[2+fraca^2N_o]^-1$$
Where the probability density function of the r.v. is $f(u)=frac1mucdotexp^frac-umu$ with $mu$ is the mean. My question is how to calculate the error probability since $x$ and $y$ have different mean values?
To my knowledge, I think we can calculate $mathbbPy>0$ with $int_0^infty
frac1N_ocdotexp^frac-uN_odu$ (please correct me if I am wrong). Thus, how do I extend it to two r.v. with different mean values?





probability-distributions density-function





share|cite|improve this question












share|cite|improve this question




share|cite|improve this question








edited Jul 22 at 6:13
























asked Jul 22 at 2:27









M.A.N

1007




1007











  • I don't understand -- you say "We can calculate ..." and give the result, but then you seem to ask how to obtain it. Is this a result from the book? Who's the "we" who can calculate it?
    – joriki
    Jul 22 at 10:41











  • I think this post will answer your question
    – David M.
    Jul 22 at 15:34










  • Thanks for the help guys. I have found a relevant link. I apologize for the repeated question. The answer can be found here: math.stackexchange.com/questions/1805587/…
    – M.A.N
    Jul 23 at 1:02
















  • I don't understand -- you say "We can calculate ..." and give the result, but then you seem to ask how to obtain it. Is this a result from the book? Who's the "we" who can calculate it?
    – joriki
    Jul 22 at 10:41











  • I think this post will answer your question
    – David M.
    Jul 22 at 15:34










  • Thanks for the help guys. I have found a relevant link. I apologize for the repeated question. The answer can be found here: math.stackexchange.com/questions/1805587/…
    – M.A.N
    Jul 23 at 1:02















I don't understand -- you say "We can calculate ..." and give the result, but then you seem to ask how to obtain it. Is this a result from the book? Who's the "we" who can calculate it?
– joriki
Jul 22 at 10:41





I don't understand -- you say "We can calculate ..." and give the result, but then you seem to ask how to obtain it. Is this a result from the book? Who's the "we" who can calculate it?
– joriki
Jul 22 at 10:41













I think this post will answer your question
– David M.
Jul 22 at 15:34




I think this post will answer your question
– David M.
Jul 22 at 15:34












Thanks for the help guys. I have found a relevant link. I apologize for the repeated question. The answer can be found here: math.stackexchange.com/questions/1805587/…
– M.A.N
Jul 23 at 1:02




Thanks for the help guys. I have found a relevant link. I apologize for the repeated question. The answer can be found here: math.stackexchange.com/questions/1805587/…
– M.A.N
Jul 23 at 1:02















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