Tight upper bound on the expectation of a concave function
$begingroup$
N is a random variable whose sample space is $[0,infty)$. I have an expression in terms of the expectation of this variable and I want to find a tight upper bound on the whole expression. The expression is as below:
$1-gamma E[frac{(N+1)}{(1+bcdot N+gamma cdot N+b+gamma)}]$ where $gamma >0$ and $b>0$
I don't know the distribution of N but assuming that I know the $E[N]$, can a tight bound be established on the above expression? If so, can you please refer me to theorems/inequalities that would help me find it?
I looked at Jensen's inequality but it provides a lower bound since the expression over which the expectation is being taken is concave.
(This expression is actually a bound on the probability mass for a subset of the sample space but I don't know the distribution.)
PS: I have asked this question on other platforms but haven't received any answers.
expected-value
edited Dec 6 '18 at 5:37
Tianlalu
3,08121038
asked Dec 6 '18 at 4:38
gagansogaganso
155211
$endgroup$
add a comment |
$begingroup$
N is a random variable whose sample space is $[0,infty)$. I have an expression in terms of the expectation of this variable and I want to find a tight upper bound on the whole expression. The expression is as below:
$1-gamma E[frac{(N+1)}{(1+bcdot N+gamma cdot N+b+gamma)}]$ where $gamma >0$ and $b>0$
I don't know the distribution of N but assuming that I know the $E[N]$, can a tight bound be established on the above expression? If so, can you please refer me to theorems/inequalities that would help me find it?
I looked at Jensen's inequality but it provides a lower bound since the expression over which the expectation is being taken is concave.
(This expression is actually a bound on the probability mass for a subset of the sample space but I don't know the distribution.)
PS: I have asked this question on other platforms but haven't received any answers.
expected-value
edited Dec 6 '18 at 5:37
Tianlalu
3,08121038
asked Dec 6 '18 at 4:38
gagansogaganso
155211
$endgroup$
add a comment |
$begingroup$
N is a random variable whose sample space is $[0,infty)$. I have an expression in terms of the expectation of this variable and I want to find a tight upper bound on the whole expression. The expression is as below:
$1-gamma E[frac{(N+1)}{(1+bcdot N+gamma cdot N+b+gamma)}]$ where $gamma >0$ and $b>0$
I don't know the distribution of N but assuming that I know the $E[N]$, can a tight bound be established on the above expression? If so, can you please refer me to theorems/inequalities that would help me find it?
I looked at Jensen's inequality but it provides a lower bound since the expression over which the expectation is being taken is concave.
(This expression is actually a bound on the probability mass for a subset of the sample space but I don't know the distribution.)
PS: I have asked this question on other platforms but haven't received any answers.
expected-value
edited Dec 6 '18 at 5:37
Tianlalu
3,08121038
asked Dec 6 '18 at 4:38
gagansogaganso
155211
$endgroup$
N is a random variable whose sample space is $[0,infty)$. I have an expression in terms of the expectation of this variable and I want to find a tight upper bound on the whole expression. The expression is as below:
$1-gamma E[frac{(N+1)}{(1+bcdot N+gamma cdot N+b+gamma)}]$ where $gamma >0$ and $b>0$
I don't know the distribution of N but assuming that I know the $E[N]$, can a tight bound be established on the above expression? If so, can you please refer me to theorems/inequalities that would help me find it?
I looked at Jensen's inequality but it provides a lower bound since the expression over which the expectation is being taken is concave.
(This expression is actually a bound on the probability mass for a subset of the sample space but I don't know the distribution.)
PS: I have asked this question on other platforms but haven't received any answers.
expected-value
expected-value
edited Dec 6 '18 at 5:37
Tianlalu
3,08121038
asked Dec 6 '18 at 4:38
gagansogaganso
155211
edited Dec 6 '18 at 5:37
Tianlalu
3,08121038
asked Dec 6 '18 at 4:38
gagansogaganso
155211
edited Dec 6 '18 at 5:37
Tianlalu
3,08121038
edited Dec 6 '18 at 5:37
Tianlalu
3,08121038
edited Dec 6 '18 at 5:37
Tianlalu
3,08121038
3,08121038
asked Dec 6 '18 at 4:38
gagansogaganso
155211
asked Dec 6 '18 at 4:38
gagansogaganso
155211
asked Dec 6 '18 at 4:38
gagansogaganso
155211
155211
add a comment |
add a comment |
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